Explanation:
Okay, so I need to solve the inequality 1 divided by (x minus 5) is greater than zero. Let me write that down first: 1/(x - 5) > 0. Alright, so I need to find all the x values that make this true. Hmm, let me think about how to approach this.

First, I remember that when dealing with rational inequalities, the key is to determine where the expression is positive or negative. Since the numerator here is 1, which is always positive, the sign of the entire expression depends on the denominator. Because if I have a positive number divided by something, the result is positive if the denominator is positive, and negative if the denominator is negative. So, if I can figure out when the denominator is positive, that should give me the solution.

So let me think: 1/(x - 5) > 0. Since 1 is positive, the denominator (x - 5) must also be positive for the whole expression to be positive. Therefore, I need to solve the inequality x - 5 > 0. That seems straightforward. Solving x - 5 > 0 would be x > 5. So does that mean the solution is all real numbers greater than 5?

Wait, but hold on. I should check if there are any restrictions on the domain. The original expression has a denominator, so x cannot be 5 because that would make the denominator zero, which is undefined. So x = 5 is not allowed. So the domain is all real numbers except x = 5. Therefore, the solution to the inequality is all x such that x > 5. But let me verify that.

Let me take a test value in the interval x > 5, say x = 6. Plugging into the expression, 1/(6 - 5) = 1/1 = 1, which is indeed greater than 0. What about a test value less than 5? Let's try x = 4. Then, 1/(4 - 5) = 1/(-1) = -1, which is less than 0. So that's not part of the solution. What about x = 5? That's undefined, so it's not included. So seems like only x > 5 works.

Wait, but let me think again. Since the inequality is strict (> 0), not greater than or equal to, so we don't include the point where the expression is zero. But in this case, the expression can never be zero because the numerator is 1, which is never zero. So there's no solution where 1/(x - 5) equals zero. Therefore, the only critical point is x = 5 where the expression is undefined.

So, putting that all together, the solution is x > 5. In interval notation, that would be (5, ∞). Let me just make sure I didn't miss anything. Sometimes with rational expressions, there can be more critical points, but here the numerator is 1, so no other critical points. The only point where the expression changes sign is at x = 5. So the expression is positive when x is greater than 5 and negative when x is less than 5. Therefore, the solution set is indeed all real numbers greater than 5.

Alternatively, maybe I can solve it algebraically by considering the inequality 1/(x - 5) > 0. Multiplying both sides by (x - 5). Wait, but I need to be careful here because multiplying both sides of an inequality by a negative number reverses the inequality sign. However, since I don't know the sign of (x - 5), I can't just multiply both sides without considering cases. Let me try that approach.

Case 1: Suppose x - 5 > 0. Then multiplying both sides by (x - 5) does not change the inequality direction. So 1 > 0*(x - 5), which is 1 > 0. Which is true. Therefore, in this case, the inequality holds when x - 5 > 0, so x > 5.

Case 2: Suppose x - 5 < 0. Then multiplying both sides by (x - 5), which is negative, reverses the inequality sign. So the original inequality 1/(x - 5) > 0 becomes 1 < 0*(x - 5), which simplifies to 1 < 0. Which is false. Therefore, there are no solutions in this case.

Case 3: If x - 5 = 0, then the expression is undefined, so there's no solution here either.

So combining the cases, the solution is x > 5. So again, the same conclusion. Therefore, in interval notation, it's (5, ∞).

Wait, but let me check if there's another method. Maybe using a sign chart. Let me try that too.

The critical point is x = 5. So we can divide the number line into intervals based on this critical point. The intervals are (-∞, 5) and (5, ∞).

For the interval (-∞, 5), pick a test point, say x = 0. Then, x - 5 = -5, so 1/(-5) = -1/5 < 0. So the expression is negative here.

For the interval (5, ∞), pick x = 6, which gives 1/(6 - 5) = 1 > 0. So positive here.

Therefore, the expression is positive only on (5, ∞). Since the inequality is strictly greater than zero, we don't include x = 5. So the same result.

So all methods point to the solution being (5, ∞). Therefore, I think that's the correct answer.

But just to make sure I haven't overlooked something, let me think if there are any other points where the expression could be positive. For example, if the numerator was something else, maybe there could be more intervals, but here numerator is 1, which is positive. So the sign of the whole expression depends solely on the denominator. Therefore, the only interval where denominator is positive is x > 5. So yeah, that's the solution.

Alternatively, if I graph the function f(x) = 1/(x - 5), it's a hyperbola with vertical asymptote at x = 5. The graph is in the first and fourth quadrants. So for x > 5, it's positive, and for x < 5, it's negative. So the graph is positive when x > 5, so the solution is x > 5. That's consistent with previous results.

Therefore, I think I can be confident that the solution is all real numbers greater than 5, which in interval notation is (5, ∞).

**Final Answer**
The solution to the inequality is \boxed{(5, \infty)}.

To solve the inequality

$$
\frac{1}{x - 5} > 0,
$$

we begin by analyzing the structure of the expression.

---

### Step 1: Understand the Expression

The left-hand side of the inequality is a **rational expression**, with the numerator being **1** (a constant, always positive), and the denominator being $ x - 5 $. The expression is defined **only when the denominator is not zero**, i.e., when $ x \ne 5 $.

---

### Step 2: Determine When the Expression is Positive

Since the numerator is **positive** (1), the entire expression will be **positive** when the denominator is also **positive**, and **negative** when the denominator is **negative**.

So, the inequality

$$
\frac{1}{x - 5} > 0
$$

is true **if and only if** the denominator $ x - 5 > 0 $.

---

### Step 3: Solve the Inequality

$$
x - 5 > 0 \Rightarrow x > 5
$$

So, the inequality is satisfied for all $ x > 5 $. However, recall that at $ x = 5 $, the expression is **undefined** because the denominator becomes zero. Therefore, we **exclude** $ x = 5 $ from the solution set.

---

### Step 4: Write the Solution in Interval Notation

The solution set consists of all real numbers greater than 5, but excluding 5 itself. This corresponds to the interval:

$$
(5, \infty)
$$

---

### Final Answer

$$
\boxed{(5, \infty)}
$$

[Assistant]
Explanation:
Okay, so I need to solve the inequality 1 divided by (x minus 5) is greater than zero. Let me write that down first: 1/(x - 5) > 0. Alright, so I need to find all the x values that make this true. Hmm, let me think about how to approach this.

First, I remember that when dealing with rational inequalities, the key is to determine where the expression is positive or negative. Since the numerator here is 1, which is always positive, the sign of the entire expression depends on the denominator. Because if I have a positive number divided by something, the result is positive if the denominator is positive, and negative if the denominator is negative. So, if I can figure out when the denominator is positive, that should give me the solution.

So let me think: 1/(x - 5) > 0. Since 1 is positive, the denominator (x - 5) must also be positive for the whole expression to be positive. Therefore, I need to solve the inequality x - 5 > 0. That seems straightforward. Solving x - 5 > 0 would be x > 5. So does that mean the solution is all real numbers greater than 5?

Wait, but hold on. I should check if there are any restrictions on the domain. The original expression has a denominator, so x cannot be 5 because that would make the denominator zero, which is undefined. So x = 5 is not allowed. So the domain is all real numbers except x = 5. Therefore, the solution to the inequality is all x such that x > 5. But let me verify that.

Let me take a test value in the interval x > 5, say x = 6. Plugging into the expression, 1/(6 - 5) = 1/1 = 1, which is indeed greater than 0. What about a test value less than 5? Let's try x = 4. Then, 1/(4 - 5) = 1/(-1) = -1, which is less than 0. So that's not part of the solution. What about x = 5? That's undefined, so it's not included. So seems like only x > 5 works.

Wait, but let me think again. Since the inequality is strict (> 0), not greater than or equal to, so we don't include the point where the expression is zero. But in this case, the expression can never be zero because the numerator is 1, which is never zero. So there's no solution where 1/(x - 5) equals zero. Therefore, the only critical point is x = 5 where the expression is undefined.

So, putting that all together, the solution is x > 5. In interval notation, that would be (5, ∞). Let me just make sure I didn't miss anything. Sometimes with rational expressions, there can be more critical points, but here the numerator is 1, so no other critical points. The only point where the expression changes sign is at x = 5. So the expression is positive when x is greater than 5 and negative when x is less than 5. Therefore, the solution set is indeed all real numbers greater than 5.

Alternatively, maybe I can solve it algebraically by considering the inequality 1/(x - 5) > 0. Multiplying both sides by (x - 5). Wait, but I need to be careful here because multiplying both sides of an inequality by a negative number reverses the inequality sign. However, since I don't know the sign of (x - 5), I can't just multiply both sides without considering cases. Let me try that approach.

Case 1: Suppose x - 5 > 0. Then multiplying both sides by (x - 5) does not change the inequality direction. So 1 > 0*(x - 5), which is 1 > 0. Which is true. Therefore, in this case, the inequality holds when x - 5 > 0, so x > 5.

Case 2: Suppose x - 5 < 0. Then multiplying both sides by (x - 5), which is negative, reverses the inequality sign. So the original inequality 1/(x - 5) > 0 becomes 1 < 0*(x - 5), which simplifies to 1 < 0. Which is false. Therefore, there are no solutions in this case.

Case 3: If x - 5 = 0, then the expression is undefined, so there's no solution here either.

So combining the cases, the solution is x > 5. So again, the same conclusion. Therefore, in interval notation, it's (5, ∞).

Wait, but let me check if there's another method. Maybe using a sign chart. Let me try that too.

The critical point is x = 5. So we can divide the number line into intervals based on this critical point. The intervals are (-∞, 5) and (5, ∞).

For the interval (-∞, 5), pick a test point, say x = 0. Then, x - 5 = -5, so 1/(-5) = -1/5 < 0. So the expression is negative here.

For the interval (5, ∞), pick x = 6, which gives 1/(6 - 5) = 1 > 0. So positive here.

Therefore, the expression is positive only on (5, ∞). Since the inequality is strictly greater than zero, we don't include x = 5. So the same result.

So all methods point to the solution being (5, ∞). Therefore, I think that's the correct answer.

But just to make sure I haven't overlooked something, let me think if there are any other points where the expression could be positive. For example, if the numerator was something else, maybe there could be more intervals, but here numerator is 1, which is positive. So the sign of the whole expression depends solely on the denominator. Therefore, the only interval where denominator is positive is x > 5. So yeah, that's the solution.

Alternatively, if I graph the function f(x) = 1/(x - 5), it's a hyperbola with vertical asymptote at x = 5. The graph is in the first and fourth quadrants. So for x > 5, it's positive, and for x < 5, it's negative. So the graph is positive when x > 5, so the solution is x > 5. That's consistent with previous results.

Therefore, I think I can be confident that the solution is all real numbers greater than 5, which in interval notation is (5, ∞).

**Final Answer**
The solution to the inequality is \boxed{(5, \infty)}.

To solve the inequality

$$
\frac{1}{x - 5} > 0,
$$

we begin by analyzing the structure of the expression.

---

### Step 1: Understand the Expression

The left-hand side of the inequality is a **rational expression**, with the numerator being **1** (a constant, always positive), and the denominator being $ x - 5 $. The expression is defined **only when the denominator is not zero**, i.e., when $ x \ne 5 $.

---

### Step 2: Determine When the Expression is Positive

Since the numerator is **positive** (1), the entire expression will be **positive** when the denominator is also **positive**, and **negative** when the denominator is **negative**.

So, the inequality

$$
\frac{1}{x - 5} > 0
$$

is true **if and only if** the denominator $ x - 5 > 0 $.

---

### Step 3: Solve the Inequality

$$
x - 5 > 0 \Rightarrow x > 5
$$

So, the inequality is satisfied for all $ x > 5 $. However, recall that at $ x = 5 $, the expression is **undefined** because the denominator becomes zero. Therefore, we **exclude** $ x = 5 $ from the solution set.

---

### Step 4: Write the Solution in Interval Notation

The solution set consists of all real numbers greater than 5, but excluding 5 itself. This corresponds to the interval:

$$
(5, \infty)
$$

---

### Final Answer

$$
\boxed{(5, \infty)}
$$

Solution: If $x < 5,$ then $\frac{1}{x - 5} < 0,$ and if $x > 5,$ then $\frac{1}{x - 5} > 0,$ so the solution is $x \in \boxed{(5,\infty)}.$


Explanation:
Okay, so I need to find the value of 513 squared minus 487 squared. Hmm, let me think. I remember there's a formula for the difference of squares, right? Like, a squared minus b squared equals (a minus b) times (a plus b). Let me check if I remember that correctly. Yeah, I think that's right. So maybe I can use that here to make the calculation easier instead of calculating each square separately, which might be time-consuming and prone to errors.

Alright, so if I let a be 513 and b be 487, then according to the formula, 513² - 487² should be equal to (513 - 487) multiplied by (513 + 487). Let me write that down:

513² - 487² = (513 - 487)(513 + 487)

Okay, now I need to compute both (513 - 487) and (513 + 487) first. Let me do the subtraction first. 513 minus 487. Let me calculate that. 513 minus 400 is 113, then minus 87 more. Wait, maybe it's easier to subtract directly. Let me write it out:

513
-487
------
Starting from the rightmost digit: 3 - 7. Oh, can't do that, so I need to borrow. The 3 becomes 13, and the 1 in the tens place becomes 0. Then 13 - 7 is 6. Next, the tens place: 0 - 8. Again, need to borrow. The 5 in the hundreds place becomes 4, and the 0 becomes 10. Then 10 - 8 is 2. Finally, the hundreds place: 4 - 4 is 0. So the result is 26. Wait, let me check again.

Wait, 513 - 487. Let me think of it as 513 - 400 - 87. 513 - 400 is 113. Then 113 - 87. 113 - 80 is 33, then minus 7 is 26. Yes, that's correct. So 513 - 487 is 26.

Now the addition part: 513 + 487. Let me add those numbers. 500 + 400 is 900, 13 + 87 is 100. So total is 900 + 100 = 1000. Wait, that's nice! So 513 + 487 equals 1000? Let me verify. 513 + 487. Adding units: 3 + 7 = 10. Tens: 1 + 8 = 9, plus the carried 1 is 10. Hundreds: 5 + 4 = 9, plus the carried 1 is 10. So it's 1000. Wow, that's a neat number. So the sum is 1000.

So putting it back into the formula, the difference of squares is (26)(1000). Multiplying those together. 26 times 1000 is 26,000. Wait, so is the answer 26,000? Let me think again to make sure I didn't make a mistake.

Let me recap. The difference of squares formula is correct. Then I calculated 513 - 487 as 26. Let me check that again. 487 + 26 = 513? Let's add 487 + 20 = 507, then +6 = 513. Yes, that's right. And 513 + 487 = 1000, which seems correct. 513 + 487: 500 + 400 = 900, 13 + 87 = 100, so total 1000. So multiplying 26 * 1000 gives 26,000. That seems straightforward.

Alternatively, if I wanted to check without using the formula, maybe I could compute both squares and subtract. Let me try that as a verification.

First, compute 513 squared. Hmm, 513 squared. Let me recall that (a + b)^2 = a² + 2ab + b². Maybe I can break down 513 into 500 + 13. So (500 + 13)^2 = 500² + 2*500*13 + 13².

Calculating each term:

500² is 250,000.

2*500*13 is 1000*13 = 13,000.

13² is 169.

Adding them up: 250,000 + 13,000 = 263,000; 263,000 + 169 = 263,169. So 513² is 263,169.

Now, compute 487 squared. Let me do the same. Maybe express 487 as 500 - 13. Then (500 - 13)^2 = 500² - 2*500*13 + 13².

Calculating:

500² is still 250,000.

2*500*13 is again 13,000.

13² is 169.

So (500 - 13)^2 = 250,000 - 13,000 + 169.

Calculating that: 250,000 - 13,000 = 237,000; 237,000 + 169 = 237,169. Therefore, 487² is 237,169.

Now, subtracting the two results: 263,169 - 237,169. Let me compute that. 263,169 minus 237,169. Subtracting the last four digits: 169 - 169 = 0. Then subtracting the remaining parts: 263,000 - 237,000 = 26,000. So yes, it's 26,000. That matches the previous result. So that confirms that the answer is indeed 26,000.

Therefore, using both the difference of squares formula and direct calculation, I arrive at the same answer. That gives me confidence that 26,000 is the correct value for 513² - 487².

Just to recap, using the formula was much quicker and less error-prone. Instead of dealing with large numbers and potential calculation mistakes, breaking it down using the formula simplified the problem significantly. It's always good to verify with another method when possible, especially with such large numbers, to ensure accuracy.

Another way to think about it: since 513 and 487 are both close to 500, maybe there's some symmetry or another approach. Let me see. Let me consider that 513 is 500 + 13, and 487 is 500 - 13. Therefore, their squares would be (500 + 13)^2 and (500 - 13)^2. Then, subtracting them, which is exactly what I did earlier. So the difference is [(500 + 13)^2 - (500 - 13)^2], which can also be seen as a difference of squares, which is [ (500 + 13) - (500 - 13) ] * [ (500 + 13) + (500 - 13) ].

Calculating the first part: (500 + 13) - (500 - 13) = 500 + 13 - 500 + 13 = 26. The second part: (500 + 13) + (500 - 13) = 500 + 13 + 500 - 13 = 1000. So again, 26 * 1000 = 26,000. So that's another way to see it.

Therefore, regardless of the method, the answer is consistent. Therefore, I can be confident that 26,000 is the correct value for 513 squared minus 487 squared.

**Final Answer**
The value of $513^2 - 487^2$ is \boxed{26000}.

To evaluate the expression $ 513^2 - 487^2 $, we can make use of the **difference of squares formula**:

$$
a^2 - b^2 = (a - b)(a + b)
$$

---

### Step 1: Apply the Formula

Let:
- $ a = 513 $
- $ b = 487 $

Then:

$$
513^2 - 487^2 = (513 - 487)(513 + 487)
$$

---

### Step 2: Compute $ 513 - 487 $

$$
513 - 487 = 26
$$

---

### Step 3: Compute $ 513 + 487 $

$$
513 + 487 = 1000
$$

---

### Step 4: Multiply the Results

$$
(513 - 487)(513 + 487) = 26 \times 1000 = 26000
$$

---

### Final Answer

$$
\boxed{26000}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of 513 squared minus 487 squared. Hmm, let me think. I remember there's a formula for the difference of squares, right? Like, a squared minus b squared equals (a minus b) times (a plus b). Let me check if I remember that correctly. Yeah, I think that's right. So maybe I can use that here to make the calculation easier instead of calculating each square separately, which might be time-consuming and prone to errors.

Alright, so if I let a be 513 and b be 487, then according to the formula, 513² - 487² should be equal to (513 - 487) multiplied by (513 + 487). Let me write that down:

513² - 487² = (513 - 487)(513 + 487)

Okay, now I need to compute both (513 - 487) and (513 + 487) first. Let me do the subtraction first. 513 minus 487. Let me calculate that. 513 minus 400 is 113, then minus 87 more. Wait, maybe it's easier to subtract directly. Let me write it out:

513
-487
------
Starting from the rightmost digit: 3 - 7. Oh, can't do that, so I need to borrow. The 3 becomes 13, and the 1 in the tens place becomes 0. Then 13 - 7 is 6. Next, the tens place: 0 - 8. Again, need to borrow. The 5 in the hundreds place becomes 4, and the 0 becomes 10. Then 10 - 8 is 2. Finally, the hundreds place: 4 - 4 is 0. So the result is 26. Wait, let me check again.

Wait, 513 - 487. Let me think of it as 513 - 400 - 87. 513 - 400 is 113. Then 113 - 87. 113 - 80 is 33, then minus 7 is 26. Yes, that's correct. So 513 - 487 is 26.

Now the addition part: 513 + 487. Let me add those numbers. 500 + 400 is 900, 13 + 87 is 100. So total is 900 + 100 = 1000. Wait, that's nice! So 513 + 487 equals 1000? Let me verify. 513 + 487. Adding units: 3 + 7 = 10. Tens: 1 + 8 = 9, plus the carried 1 is 10. Hundreds: 5 + 4 = 9, plus the carried 1 is 10. So it's 1000. Wow, that's a neat number. So the sum is 1000.

So putting it back into the formula, the difference of squares is (26)(1000). Multiplying those together. 26 times 1000 is 26,000. Wait, so is the answer 26,000? Let me think again to make sure I didn't make a mistake.

Let me recap. The difference of squares formula is correct. Then I calculated 513 - 487 as 26. Let me check that again. 487 + 26 = 513? Let's add 487 + 20 = 507, then +6 = 513. Yes, that's right. And 513 + 487 = 1000, which seems correct. 513 + 487: 500 + 400 = 900, 13 + 87 = 100, so total 1000. So multiplying 26 * 1000 gives 26,000. That seems straightforward.

Alternatively, if I wanted to check without using the formula, maybe I could compute both squares and subtract. Let me try that as a verification.

First, compute 513 squared. Hmm, 513 squared. Let me recall that (a + b)^2 = a² + 2ab + b². Maybe I can break down 513 into 500 + 13. So (500 + 13)^2 = 500² + 2*500*13 + 13².

Calculating each term:

500² is 250,000.

2*500*13 is 1000*13 = 13,000.

13² is 169.

Adding them up: 250,000 + 13,000 = 263,000; 263,000 + 169 = 263,169. So 513² is 263,169.

Now, compute 487 squared. Let me do the same. Maybe express 487 as 500 - 13. Then (500 - 13)^2 = 500² - 2*500*13 + 13².

Calculating:

500² is still 250,000.

2*500*13 is again 13,000.

13² is 169.

So (500 - 13)^2 = 250,000 - 13,000 + 169.

Calculating that: 250,000 - 13,000 = 237,000; 237,000 + 169 = 237,169. Therefore, 487² is 237,169.

Now, subtracting the two results: 263,169 - 237,169. Let me compute that. 263,169 minus 237,169. Subtracting the last four digits: 169 - 169 = 0. Then subtracting the remaining parts: 263,000 - 237,000 = 26,000. So yes, it's 26,000. That matches the previous result. So that confirms that the answer is indeed 26,000.

Therefore, using both the difference of squares formula and direct calculation, I arrive at the same answer. That gives me confidence that 26,000 is the correct value for 513² - 487².

Just to recap, using the formula was much quicker and less error-prone. Instead of dealing with large numbers and potential calculation mistakes, breaking it down using the formula simplified the problem significantly. It's always good to verify with another method when possible, especially with such large numbers, to ensure accuracy.

Another way to think about it: since 513 and 487 are both close to 500, maybe there's some symmetry or another approach. Let me see. Let me consider that 513 is 500 + 13, and 487 is 500 - 13. Therefore, their squares would be (500 + 13)^2 and (500 - 13)^2. Then, subtracting them, which is exactly what I did earlier. So the difference is [(500 + 13)^2 - (500 - 13)^2], which can also be seen as a difference of squares, which is [ (500 + 13) - (500 - 13) ] * [ (500 + 13) + (500 - 13) ].

Calculating the first part: (500 + 13) - (500 - 13) = 500 + 13 - 500 + 13 = 26. The second part: (500 + 13) + (500 - 13) = 500 + 13 + 500 - 13 = 1000. So again, 26 * 1000 = 26,000. So that's another way to see it.

Therefore, regardless of the method, the answer is consistent. Therefore, I can be confident that 26,000 is the correct value for 513 squared minus 487 squared.

**Final Answer**
The value of $513^2 - 487^2$ is \boxed{26000}.

To evaluate the expression $ 513^2 - 487^2 $, we can make use of the **difference of squares formula**:

$$
a^2 - b^2 = (a - b)(a + b)
$$

---

### Step 1: Apply the Formula

Let:
- $ a = 513 $
- $ b = 487 $

Then:

$$
513^2 - 487^2 = (513 - 487)(513 + 487)
$$

---

### Step 2: Compute $ 513 - 487 $

$$
513 - 487 = 26
$$

---

### Step 3: Compute $ 513 + 487 $

$$
513 + 487 = 1000
$$

---

### Step 4: Multiply the Results

$$
(513 - 487)(513 + 487) = 26 \times 1000 = 26000
$$

---

### Final Answer

$$
\boxed{26000}
$$

Solution: We note that this is a difference of squares, so $513^2 - 487^2 = (513+487)(513-487) = (1000)(26) = \boxed{26000}$.


Explanation:
Okay, let me try to figure out this probability problem. So, there are two candidates for president: Dan and Donald. And two candidates for vice president: Freddie and Bernie. Each candidate has a 50% chance of winning their respective positions. The question is asking for the probability that both Dan and Freddie win. Hmm, okay.

First, let me make sure I understand the problem correctly. There are two separate elections here: one for president and one for vice president. Each position has two candidates, and each candidate has an equal chance of winning. So, for president, Dan and Donald each have a 50% chance. Similarly, for vice president, Freddie and Bernie each have a 50% chance. The question is about the probability that both Dan (president) and Freddie (vice president) win their respective positions.

So, since these are independent events, right? The outcome of the presidential election shouldn't affect the vice presidential election, and vice versa. So, the probability that both Dan and Freddie win should be the product of their individual probabilities.

Let me verify if that makes sense. For independent events, the probability of both occurring is the product of their individual probabilities. So, if Dan has a 50% chance of winning president, and Freddie has a 50% chance of winning vice president, then multiplying those together should give the probability that both happen. So, 0.5 * 0.5 = 0.25. Which is 1/4. So, is the answer 1/4?

Wait, but maybe I need to consider all possible outcomes? Let me think again. Maybe I should list all possible outcomes and then count how many of them satisfy both Dan and Freddie winning.

So, for president, there are two possibilities: Dan wins or Donald wins. For vice president, two possibilities: Freddie wins or Bernie wins. Therefore, the total number of possible outcomes is 2 (president) * 2 (vice president) = 4 possible outcomes. Let me list them:

1. Dan wins president, Freddie wins vice president.

2. Dan wins president, Bernie wins vice president.

3. Donald wins president, Freddie wins vice president.

4. Donald wins president, Bernie wins vice president.

Each of these outcomes should be equally likely, since each candidate has a 50% chance. So, each outcome has a probability of 0.25. Therefore, the first outcome (Dan and Freddie both winning) is one of the four, so the probability is 1/4.

So that's the same as before. So, 1/4 is the answer. But let me check if that's correct.

But wait, the problem says "each candidate has a 50% chance of winning." So, does that mean that each candidate has a 50% chance regardless of the other candidates? For instance, for president, Dan and Donald each have 50%, so that's straightforward. Similarly for vice president, Freddie and Bernie each have 50%. Therefore, since these are separate elections, the results are independent. Therefore, the combined probability is indeed 1/2 * 1/2 = 1/4.

But maybe there's a trick here? Sometimes problems can be more complicated. Let me make sure. Is there any possibility that the election outcomes are dependent? For example, if the winner of one affects the other? But the problem doesn't mention any such dependencies. It just says each candidate has a 50% chance of winning. So, I think it's safe to assume independence here.

Alternatively, if the problem had said that the candidates are running in the same election, maybe with some dependencies, but here it's two separate positions. So, it's two separate elections, each with two candidates. Therefore, independent.

Alternatively, maybe the problem is in how the positions are structured. For example, maybe the president and vice president are elected together, so there's some interaction? But the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, separate positions. Therefore, each election is independent.

Alternatively, perhaps the problem is in the interpretation of "each candidate has a 50% chance of winning." Maybe that's not meaning that each candidate has a 50% chance, but that each candidate has an equal chance, which would be 50% each, since there are two candidates. So, that seems to be the case.

Alternatively, if there were more candidates, the probability might be different, but since there are only two candidates per position, each has 50%.

Therefore, the answer is 1/4.

Wait, but maybe the problem is not about independent elections? Let me think again. For example, in some cases, people might think that if Dan is president, maybe that affects Freddie's chances? But the problem says each candidate has 50% chance. So, maybe it's given that each candidate has 50% chance regardless of the others? So, even if there are other candidates, but here there are only two. So, maybe the answer is indeed 1/4.

Alternatively, maybe there is a different way to model the problem. Let's think of it as each position being a separate Bernoulli trial with two outcomes, each with probability 1/2. Then, the joint probability is the product.

Alternatively, if we think about the sample space, which has four outcomes, each with equal probability, and only one of those outcomes has Dan and Freddie both winning, so 1/4.

But let me think if there is another angle. For example, sometimes in probability problems, people get confused between permutations and combinations. But here, since the positions are distinct, the outcomes are distinct. So, the four outcomes are distinct and equally likely.

Another way: if you consider that for the president, Dan has a 1/2 chance, and for the vice president, Freddie has a 1/2 chance. Since the events are independent, multiply them. So 1/2 * 1/2 = 1/4.

Alternatively, if the problem had said that there are four candidates for president and vice president, but here it's two separate positions with two candidates each.

Wait, but maybe the problem is in the wording "each candidate has a 50% chance of winning." So, does that mean that for each candidate, regardless of the position, they have 50% chance? But that would be strange. For example, if there are two candidates for president, then each has 50% chance. Similarly, for vice president. But if there were more candidates, it might be different. So, given that, the answer is 1/4.

Alternatively, maybe the problem is that when they say "each candidate has a 50% chance of winning," maybe they mean that each candidate has 50% chance of winning their own position? Which is what we assumed. So, Dan and Donald each have 50% chance for president, Freddie and Bernie each have 50% chance for vice president. So, independent.

Alternatively, perhaps the problem is in the fact that there are two separate elections, but maybe there's some overlap in the candidates? Wait, the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, the candidates for president are Dan and Donald, and the candidates for vice president are Freddie and Bernie. So, no overlap. Therefore, there's no overlap between the candidates for president and vice president. Therefore, their winning is independent.

Therefore, combining the probabilities is straightforward.

Alternatively, if there were overlap, say, if Freddie was also running for president, then the events might be dependent. But in this case, they are separate.

So, given all that, I think the answer is 1/4.

But let me check once again. Suppose that the probability of Dan winning is 1/2, and the probability of Freddie winning is 1/2, and they are independent. Then, the probability that both happen is 1/2 * 1/2 = 1/4.

Alternatively, if they were not independent, then we would need more information. But here, we are told that each candidate has a 50% chance, which suggests independence.

Alternatively, maybe the problem is testing something else. For example, maybe the question is about the probability that both Dan and Freddie are elected, but considering that there are two positions, so maybe some other way?

Wait, but the problem says "both Dan and Freddie win." Since Dan is running for president and Freddie for vice president, so they are different positions. So, it's possible for both to win. So, the question is simply asking for the probability that Dan is elected president and Freddie is elected vice president.

Therefore, since these are independent events, the probability is indeed 1/2 * 1/2 = 1/4.

Alternatively, if you think about all possible outcomes:

- President: Dan or Donald

- Vice President: Freddie or Bernie

So, the four possibilities:

1. Dan (President) and Freddie (Vice President)

2. Dan (President) and Bernie (Vice President)

3. Donald (President) and Freddie (Vice President)

4. Donald (President) and Bernie (Vice President)

Each of these has equal probability, since each candidate has 50% chance. So, each outcome has probability 1/2 * 1/2 = 1/4. Therefore, the probability that both Dan and Freddie win is 1/4.

Therefore, the answer is 1/4.

But let me check if there's a different interpretation. For example, maybe "each candidate has a 50% chance of winning" refers to all candidates collectively? But that doesn't make much sense. For example, if there are four candidates total (Dan, Donald, Freddie, Bernie), but they are running for two different positions. But the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, there are two positions, each with two candidates. So, each candidate is running for one position, so each candidate is only in one election. Therefore, the probability that Dan wins president is 50%, and Freddie wins vice president is 50%, and they are independent.

Alternatively, maybe the problem is trying to trick us into thinking that there are four candidates for two positions? But no, the problem says two candidates for president and two for vice president, so four candidates in total.

But even if there are four candidates, since they are in separate positions, the probabilities are still independent.

Alternatively, maybe the problem is considering that when you have multiple positions, there's some sort of dependency? For example, if someone is elected president, maybe they can't be elected vice president? But in this case, the candidates are different. Dan and Donald are running for president, Freddie and Bernie for vice president. So, there is no overlap, so they can't be elected to both positions. So, the president and vice president are different people.

So, in that case, the events of Dan winning president and Freddie winning vice president are independent, because the election for president doesn't affect the election for vice president.

So, yeah, that seems right. Therefore, the probability is 1/2 * 1/2 = 1/4.

Alternatively, if there was a scenario where, for instance, if the president and vice president had to be different people, but since they are different candidates, that's already taken care of.

Wait, but in this case, since the candidates for president and vice president are distinct, there's no issue. So, the president and vice president are different people. So, there's no overlap. Therefore, the events are independent.

So, yeah, I think that settles it. The answer is 1/4.

But to be thorough, let me think of another approach. Let's use probability axioms.

Let A be the event that Dan wins president. Let B be the event that Freddie wins vice president. We are to find P(A and B).

If A and B are independent, then P(A and B) = P(A) * P(B).

Given that each candidate has a 50% chance of winning, then P(A) = 1/2, P(B) = 1/2. Therefore, if they are independent, P(A and B) = 1/2 * 1/2 = 1/4.

So, the question becomes: are events A and B independent?

To check independence, we need to see if P(A and B) = P(A) * P(B). However, without additional information, we have to assume independence if the problem doesn't specify any dependence. Since the problem states that each candidate has a 50% chance, and there is no mention of any relation between the two positions, we can assume independence.

Alternatively, if we didn't assume independence, how would we calculate it? Well, we might need more information. But since the problem gives us only that each candidate has a 50% chance, and gives no other information, we must assume independence.

Therefore, the probability is 1/4.

So, after all these considerations, I believe the answer is 1/4.

**Final Answer**
The probability that both Dan and Freddie win is \boxed{\dfrac{1}{4}}.

To solve the problem, we consider two independent election events:

- **President**: Dan and Donald are the two candidates, each with a **50% chance** of winning.
- **Vice President**: Freddie and Bernie are the two candidates, each with a **50% chance** of winning.

---

### Step 1: Understand the Nature of the Events

The question asks for the probability that **both Dan (president) and Freddie (vice president)** win their respective positions.

- Because the two positions (president and vice president) are **separate**, and the outcomes of one do not affect the other, these are **independent events**.
- Therefore, we can compute the probability of both events happening by **multiplying** their individual probabilities.

---

### Step 2: Apply the Multiplication Rule

Let:

- $ P(\text{Dan wins}) = \frac{1}{2} $
- $ P(\text{Freddie wins}) = \frac{1}{2} $

Since these are independent:

$$
P(\text{Dan wins and Freddie wins}) = P(\text{Dan wins}) \times P(\text{Freddie wins}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
$$

---

### Step 3: Confirm via Outcome Enumeration

We can also confirm by listing all possible combinations of election outcomes:

1. Dan (President), Freddie (Vice President)
2. Dan (President), Bernie (Vice President)
3. Donald (President), Freddie (Vice President)
4. Donald (President), Bernie (Vice President)

Since each candidate has an equal 50% chance and the elections are independent, each of these four outcomes has an equal probability of $ \frac{1}{4} $. Only **one of these four outcomes** satisfies the condition that **both Dan and Freddie win**, confirming that the probability is:

$$
\frac{1}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{4}}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this probability problem. So, there are two candidates for president: Dan and Donald. And two candidates for vice president: Freddie and Bernie. Each candidate has a 50% chance of winning their respective positions. The question is asking for the probability that both Dan and Freddie win. Hmm, okay.

First, let me make sure I understand the problem correctly. There are two separate elections here: one for president and one for vice president. Each position has two candidates, and each candidate has an equal chance of winning. So, for president, Dan and Donald each have a 50% chance. Similarly, for vice president, Freddie and Bernie each have a 50% chance. The question is about the probability that both Dan (president) and Freddie (vice president) win their respective positions.

So, since these are independent events, right? The outcome of the presidential election shouldn't affect the vice presidential election, and vice versa. So, the probability that both Dan and Freddie win should be the product of their individual probabilities.

Let me verify if that makes sense. For independent events, the probability of both occurring is the product of their individual probabilities. So, if Dan has a 50% chance of winning president, and Freddie has a 50% chance of winning vice president, then multiplying those together should give the probability that both happen. So, 0.5 * 0.5 = 0.25. Which is 1/4. So, is the answer 1/4?

Wait, but maybe I need to consider all possible outcomes? Let me think again. Maybe I should list all possible outcomes and then count how many of them satisfy both Dan and Freddie winning.

So, for president, there are two possibilities: Dan wins or Donald wins. For vice president, two possibilities: Freddie wins or Bernie wins. Therefore, the total number of possible outcomes is 2 (president) * 2 (vice president) = 4 possible outcomes. Let me list them:

1. Dan wins president, Freddie wins vice president.

2. Dan wins president, Bernie wins vice president.

3. Donald wins president, Freddie wins vice president.

4. Donald wins president, Bernie wins vice president.

Each of these outcomes should be equally likely, since each candidate has a 50% chance. So, each outcome has a probability of 0.25. Therefore, the first outcome (Dan and Freddie both winning) is one of the four, so the probability is 1/4.

So that's the same as before. So, 1/4 is the answer. But let me check if that's correct.

But wait, the problem says "each candidate has a 50% chance of winning." So, does that mean that each candidate has a 50% chance regardless of the other candidates? For instance, for president, Dan and Donald each have 50%, so that's straightforward. Similarly for vice president, Freddie and Bernie each have 50%. Therefore, since these are separate elections, the results are independent. Therefore, the combined probability is indeed 1/2 * 1/2 = 1/4.

But maybe there's a trick here? Sometimes problems can be more complicated. Let me make sure. Is there any possibility that the election outcomes are dependent? For example, if the winner of one affects the other? But the problem doesn't mention any such dependencies. It just says each candidate has a 50% chance of winning. So, I think it's safe to assume independence here.

Alternatively, if the problem had said that the candidates are running in the same election, maybe with some dependencies, but here it's two separate positions. So, it's two separate elections, each with two candidates. Therefore, independent.

Alternatively, maybe the problem is in how the positions are structured. For example, maybe the president and vice president are elected together, so there's some interaction? But the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, separate positions. Therefore, each election is independent.

Alternatively, perhaps the problem is in the interpretation of "each candidate has a 50% chance of winning." Maybe that's not meaning that each candidate has a 50% chance, but that each candidate has an equal chance, which would be 50% each, since there are two candidates. So, that seems to be the case.

Alternatively, if there were more candidates, the probability might be different, but since there are only two candidates per position, each has 50%.

Therefore, the answer is 1/4.

Wait, but maybe the problem is not about independent elections? Let me think again. For example, in some cases, people might think that if Dan is president, maybe that affects Freddie's chances? But the problem says each candidate has 50% chance. So, maybe it's given that each candidate has 50% chance regardless of the others? So, even if there are other candidates, but here there are only two. So, maybe the answer is indeed 1/4.

Alternatively, maybe there is a different way to model the problem. Let's think of it as each position being a separate Bernoulli trial with two outcomes, each with probability 1/2. Then, the joint probability is the product.

Alternatively, if we think about the sample space, which has four outcomes, each with equal probability, and only one of those outcomes has Dan and Freddie both winning, so 1/4.

But let me think if there is another angle. For example, sometimes in probability problems, people get confused between permutations and combinations. But here, since the positions are distinct, the outcomes are distinct. So, the four outcomes are distinct and equally likely.

Another way: if you consider that for the president, Dan has a 1/2 chance, and for the vice president, Freddie has a 1/2 chance. Since the events are independent, multiply them. So 1/2 * 1/2 = 1/4.

Alternatively, if the problem had said that there are four candidates for president and vice president, but here it's two separate positions with two candidates each.

Wait, but maybe the problem is in the wording "each candidate has a 50% chance of winning." So, does that mean that for each candidate, regardless of the position, they have 50% chance? But that would be strange. For example, if there are two candidates for president, then each has 50% chance. Similarly, for vice president. But if there were more candidates, it might be different. So, given that, the answer is 1/4.

Alternatively, maybe the problem is that when they say "each candidate has a 50% chance of winning," maybe they mean that each candidate has 50% chance of winning their own position? Which is what we assumed. So, Dan and Donald each have 50% chance for president, Freddie and Bernie each have 50% chance for vice president. So, independent.

Alternatively, perhaps the problem is in the fact that there are two separate elections, but maybe there's some overlap in the candidates? Wait, the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, the candidates for president are Dan and Donald, and the candidates for vice president are Freddie and Bernie. So, no overlap. Therefore, there's no overlap between the candidates for president and vice president. Therefore, their winning is independent.

Therefore, combining the probabilities is straightforward.

Alternatively, if there were overlap, say, if Freddie was also running for president, then the events might be dependent. But in this case, they are separate.

So, given all that, I think the answer is 1/4.

But let me check once again. Suppose that the probability of Dan winning is 1/2, and the probability of Freddie winning is 1/2, and they are independent. Then, the probability that both happen is 1/2 * 1/2 = 1/4.

Alternatively, if they were not independent, then we would need more information. But here, we are told that each candidate has a 50% chance, which suggests independence.

Alternatively, maybe the problem is testing something else. For example, maybe the question is about the probability that both Dan and Freddie are elected, but considering that there are two positions, so maybe some other way?

Wait, but the problem says "both Dan and Freddie win." Since Dan is running for president and Freddie for vice president, so they are different positions. So, it's possible for both to win. So, the question is simply asking for the probability that Dan is elected president and Freddie is elected vice president.

Therefore, since these are independent events, the probability is indeed 1/2 * 1/2 = 1/4.

Alternatively, if you think about all possible outcomes:

- President: Dan or Donald

- Vice President: Freddie or Bernie

So, the four possibilities:

1. Dan (President) and Freddie (Vice President)

2. Dan (President) and Bernie (Vice President)

3. Donald (President) and Freddie (Vice President)

4. Donald (President) and Bernie (Vice President)

Each of these has equal probability, since each candidate has 50% chance. So, each outcome has probability 1/2 * 1/2 = 1/4. Therefore, the probability that both Dan and Freddie win is 1/4.

Therefore, the answer is 1/4.

But let me check if there's a different interpretation. For example, maybe "each candidate has a 50% chance of winning" refers to all candidates collectively? But that doesn't make much sense. For example, if there are four candidates total (Dan, Donald, Freddie, Bernie), but they are running for two different positions. But the problem says "two candidates, Dan and Donald, run for class president. Two other candidates, Freddie and Bernie, run for vice president." So, there are two positions, each with two candidates. So, each candidate is running for one position, so each candidate is only in one election. Therefore, the probability that Dan wins president is 50%, and Freddie wins vice president is 50%, and they are independent.

Alternatively, maybe the problem is trying to trick us into thinking that there are four candidates for two positions? But no, the problem says two candidates for president and two for vice president, so four candidates in total.

But even if there are four candidates, since they are in separate positions, the probabilities are still independent.

Alternatively, maybe the problem is considering that when you have multiple positions, there's some sort of dependency? For example, if someone is elected president, maybe they can't be elected vice president? But in this case, the candidates are different. Dan and Donald are running for president, Freddie and Bernie for vice president. So, there is no overlap, so they can't be elected to both positions. So, the president and vice president are different people.

So, in that case, the events of Dan winning president and Freddie winning vice president are independent, because the election for president doesn't affect the election for vice president.

So, yeah, that seems right. Therefore, the probability is 1/2 * 1/2 = 1/4.

Alternatively, if there was a scenario where, for instance, if the president and vice president had to be different people, but since they are different candidates, that's already taken care of.

Wait, but in this case, since the candidates for president and vice president are distinct, there's no issue. So, the president and vice president are different people. So, there's no overlap. Therefore, the events are independent.

So, yeah, I think that settles it. The answer is 1/4.

But to be thorough, let me think of another approach. Let's use probability axioms.

Let A be the event that Dan wins president. Let B be the event that Freddie wins vice president. We are to find P(A and B).

If A and B are independent, then P(A and B) = P(A) * P(B).

Given that each candidate has a 50% chance of winning, then P(A) = 1/2, P(B) = 1/2. Therefore, if they are independent, P(A and B) = 1/2 * 1/2 = 1/4.

So, the question becomes: are events A and B independent?

To check independence, we need to see if P(A and B) = P(A) * P(B). However, without additional information, we have to assume independence if the problem doesn't specify any dependence. Since the problem states that each candidate has a 50% chance, and there is no mention of any relation between the two positions, we can assume independence.

Alternatively, if we didn't assume independence, how would we calculate it? Well, we might need more information. But since the problem gives us only that each candidate has a 50% chance, and gives no other information, we must assume independence.

Therefore, the probability is 1/4.

So, after all these considerations, I believe the answer is 1/4.

**Final Answer**
The probability that both Dan and Freddie win is \boxed{\dfrac{1}{4}}.

To solve the problem, we consider two independent election events:

- **President**: Dan and Donald are the two candidates, each with a **50% chance** of winning.
- **Vice President**: Freddie and Bernie are the two candidates, each with a **50% chance** of winning.

---

### Step 1: Understand the Nature of the Events

The question asks for the probability that **both Dan (president) and Freddie (vice president)** win their respective positions.

- Because the two positions (president and vice president) are **separate**, and the outcomes of one do not affect the other, these are **independent events**.
- Therefore, we can compute the probability of both events happening by **multiplying** their individual probabilities.

---

### Step 2: Apply the Multiplication Rule

Let:

- $ P(\text{Dan wins}) = \frac{1}{2} $
- $ P(\text{Freddie wins}) = \frac{1}{2} $

Since these are independent:

$$
P(\text{Dan wins and Freddie wins}) = P(\text{Dan wins}) \times P(\text{Freddie wins}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
$$

---

### Step 3: Confirm via Outcome Enumeration

We can also confirm by listing all possible combinations of election outcomes:

1. Dan (President), Freddie (Vice President)
2. Dan (President), Bernie (Vice President)
3. Donald (President), Freddie (Vice President)
4. Donald (President), Bernie (Vice President)

Since each candidate has an equal 50% chance and the elections are independent, each of these four outcomes has an equal probability of $ \frac{1}{4} $. Only **one of these four outcomes** satisfies the condition that **both Dan and Freddie win**, confirming that the probability is:

$$
\frac{1}{4}
$$

---

### Final Answer

$$
\boxed{\dfrac{1}{4}}
$$

Solution: The probability that Dan wins is $\frac12$. The probability that Freddie wins is also $\frac12$. Therefore, the probability that both win is $\frac12 \cdot \frac12 =\boxed{\frac14}$.


Explanation:
Okay, so I need to find the dot product of two vectors. Let me recall what a dot product is. From what I remember, the dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Let me check if I got that right.

So, if I have two vectors, say, vector **a** = [a₁, a₂, a₃] and vector **b** = [b₁, b₂, b₃], then their dot product **a** · **b** should be a₁*b₁ + a₂*b₂ + a₃*b₃. Yeah, that seems familiar from my notes. Okay, so the formula is straightforward: multiply each pair of corresponding entries and sum them up.

Alright, let's apply this to the given vectors. The first vector is [-2, 0, 7], and the second one is [3, 4, -5]. Let me write them down again to make sure I have them right:

First vector:
- First component: -2
- Second component: 0
- Third component: 7

Second vector:
- First component: 3
- Second component: 4
- Third component: -5

So, according to the formula, I need to compute (-2)*3 + 0*4 + 7*(-5). Let me compute each term step by step.

Starting with the first term: (-2) multiplied by 3. Let me calculate that. Multiplying a negative number by a positive number gives a negative result. So, -2 * 3 is -6. Got that.

Next term: 0 multiplied by 4. Well, anything multiplied by zero is zero. So that term is 0. Easy enough.

Third term: 7 multiplied by -5. Hmm, 7 times 5 is 35, and since one of them is negative, the result should be negative. So that's -35.

Now, I need to add these three results together: -6 + 0 + (-35). Let me do that step by step.

First, adding -6 and 0. Well, adding zero doesn't change anything, so that's still -6. Then, adding -35 to that. So, -6 + (-35) is the same as -(6 + 35), right? Which is -41.

Wait, let me confirm that. If I have a debt of 6 dollars and then another debt of 35 dollars, my total debt is 41 dollars. So, yes, combining those two negative numbers gives me -41.

Therefore, the dot product of these two vectors is -41.

But let me double-check my calculations to make sure I didn't make any errors.

First component multiplication: -2 * 3 = -6. Correct.

Second component: 0 * 4 = 0. Definitely right.

Third component: 7 * (-5) = -35. Yep, that's correct.

Adding them up: -6 + 0 + (-35). Let's rearrange the terms: (-6) + (-35) + 0. Which is (-6 -35) + 0 = -41 + 0 = -41.

So that seems consistent.

Wait, just to make sure, is there another way to verify this? Maybe by using another method or checking the properties of dot products?

Well, the dot product is also related to the magnitudes of the vectors and the cosine of the angle between them. But since I don't know the angle here, that might not be helpful. Alternatively, maybe I can compute the magnitudes and see if the dot product makes sense?

Let me try that. Let's compute the magnitudes of both vectors.

First vector: magnitude is sqrt((-2)^2 + 0^2 + 7^2) = sqrt(4 + 0 + 49) = sqrt(53).

Second vector: magnitude is sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50).

So the product of the magnitudes is sqrt(53) * sqrt(50) = sqrt(53*50). Let me calculate that. 53*50 is 2650. So sqrt(2650).

Now, the dot product is equal to |a||b|cos(theta). So, if I compute |a||b|cos(theta), it should equal -41. But since I don't know theta, I can't directly check. However, I can check if the value -41 is a reasonable value given the magnitudes.

Let me see, sqrt(53) is approximately sqrt(49) = 7, so about 7.28. sqrt(50) is approximately 7.07. Multiplying them together gives roughly 7.28 * 7.07 ≈ 51.4.

So, if the dot product is -41, then cos(theta) would be -41 / 51.4 ≈ -0.797. Which is possible because cosine can be negative, meaning the angle is obtuse. That seems plausible.

Alternatively, if I had made a mistake in the calculation, maybe I would get a value that's not in the range of [-|a||b|, |a||b|]. Let me check if -41 is within that range.

The maximum possible value of the dot product would be |a||b| ≈ 51.4, and the minimum would be -51.4. Since -41 is greater than -51.4, it is within the valid range. So that checks out.

Therefore, my calculation seems reasonable.

Alternatively, maybe I can compute the dot product using another method. Wait, but the formula is pretty straightforward, so I don't think there's another way. But just to make sure, let me redo the calculation again.

First term: -2 * 3 = -6. Second term: 0 * 4 = 0. Third term: 7 * (-5) = -35. Total: -6 + 0 - 35 = -41. Yep, same result.

Is there a chance I mixed up the components? Let me check the original vectors again.

First vector: [-2, 0, 7]. Second vector: [3, 4, -5].

Yes, so components are:

-2, 0, 7 and 3, 4, -5.

So first components: -2 and 3. Second: 0 and 4. Third: 7 and -5. Correct.

No, I didn't mix up any components.

So all steps seem correct.

Alternatively, maybe I can think about distributive property or something? Let me try.

Suppose I consider the vectors as (-2, 0, 7) and (3, 4, -5).

Alternatively, maybe I can think of expanding some expression? Not sure.

Alternatively, if I make a mistake in sign? For example, if I thought that 7 * (-5) is positive 35? But I thought that's negative. Let me check again.

7 times negative 5: 7*5=35, and since one number is negative, the product is negative. So yes, -35.

So that term is definitely negative.

So adding -6, 0, and -35 gives me -41.

Therefore, I think my answer is correct.

I don't see any errors in my reasoning. So the dot product should be -41.

**Final Answer**
The dot product is \boxed{-41}.

To find the **dot product** of two vectors, we use the standard formula for the dot product in three-dimensional space:

$$
\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = a_1b_1 + a_2b_2 + a_3b_3
$$

---

### Step-by-step Calculation

We are given the vectors:

$$
\mathbf{a} = \begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

Now compute the dot product term by term:

1. **First components:**
   $ (-2) \times 3 = -6 $

2. **Second components:**
   $ 0 \times 4 = 0 $

3. **Third components:**
   $ 7 \times (-5) = -35 $

Now, sum the results:

$$
-6 + 0 + (-35) = -41
$$

---

### Final Answer

$$
\boxed{-41}
$$

[Assistant]
Explanation:
Okay, so I need to find the dot product of two vectors. Let me recall what a dot product is. From what I remember, the dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. Let me check if I got that right.

So, if I have two vectors, say, vector **a** = [a₁, a₂, a₃] and vector **b** = [b₁, b₂, b₃], then their dot product **a** · **b** should be a₁*b₁ + a₂*b₂ + a₃*b₃. Yeah, that seems familiar from my notes. Okay, so the formula is straightforward: multiply each pair of corresponding entries and sum them up.

Alright, let's apply this to the given vectors. The first vector is [-2, 0, 7], and the second one is [3, 4, -5]. Let me write them down again to make sure I have them right:

First vector:
- First component: -2
- Second component: 0
- Third component: 7

Second vector:
- First component: 3
- Second component: 4
- Third component: -5

So, according to the formula, I need to compute (-2)*3 + 0*4 + 7*(-5). Let me compute each term step by step.

Starting with the first term: (-2) multiplied by 3. Let me calculate that. Multiplying a negative number by a positive number gives a negative result. So, -2 * 3 is -6. Got that.

Next term: 0 multiplied by 4. Well, anything multiplied by zero is zero. So that term is 0. Easy enough.

Third term: 7 multiplied by -5. Hmm, 7 times 5 is 35, and since one of them is negative, the result should be negative. So that's -35.

Now, I need to add these three results together: -6 + 0 + (-35). Let me do that step by step.

First, adding -6 and 0. Well, adding zero doesn't change anything, so that's still -6. Then, adding -35 to that. So, -6 + (-35) is the same as -(6 + 35), right? Which is -41.

Wait, let me confirm that. If I have a debt of 6 dollars and then another debt of 35 dollars, my total debt is 41 dollars. So, yes, combining those two negative numbers gives me -41.

Therefore, the dot product of these two vectors is -41.

But let me double-check my calculations to make sure I didn't make any errors.

First component multiplication: -2 * 3 = -6. Correct.

Second component: 0 * 4 = 0. Definitely right.

Third component: 7 * (-5) = -35. Yep, that's correct.

Adding them up: -6 + 0 + (-35). Let's rearrange the terms: (-6) + (-35) + 0. Which is (-6 -35) + 0 = -41 + 0 = -41.

So that seems consistent.

Wait, just to make sure, is there another way to verify this? Maybe by using another method or checking the properties of dot products?

Well, the dot product is also related to the magnitudes of the vectors and the cosine of the angle between them. But since I don't know the angle here, that might not be helpful. Alternatively, maybe I can compute the magnitudes and see if the dot product makes sense?

Let me try that. Let's compute the magnitudes of both vectors.

First vector: magnitude is sqrt((-2)^2 + 0^2 + 7^2) = sqrt(4 + 0 + 49) = sqrt(53).

Second vector: magnitude is sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50).

So the product of the magnitudes is sqrt(53) * sqrt(50) = sqrt(53*50). Let me calculate that. 53*50 is 2650. So sqrt(2650).

Now, the dot product is equal to |a||b|cos(theta). So, if I compute |a||b|cos(theta), it should equal -41. But since I don't know theta, I can't directly check. However, I can check if the value -41 is a reasonable value given the magnitudes.

Let me see, sqrt(53) is approximately sqrt(49) = 7, so about 7.28. sqrt(50) is approximately 7.07. Multiplying them together gives roughly 7.28 * 7.07 ≈ 51.4.

So, if the dot product is -41, then cos(theta) would be -41 / 51.4 ≈ -0.797. Which is possible because cosine can be negative, meaning the angle is obtuse. That seems plausible.

Alternatively, if I had made a mistake in the calculation, maybe I would get a value that's not in the range of [-|a||b|, |a||b|]. Let me check if -41 is within that range.

The maximum possible value of the dot product would be |a||b| ≈ 51.4, and the minimum would be -51.4. Since -41 is greater than -51.4, it is within the valid range. So that checks out.

Therefore, my calculation seems reasonable.

Alternatively, maybe I can compute the dot product using another method. Wait, but the formula is pretty straightforward, so I don't think there's another way. But just to make sure, let me redo the calculation again.

First term: -2 * 3 = -6. Second term: 0 * 4 = 0. Third term: 7 * (-5) = -35. Total: -6 + 0 - 35 = -41. Yep, same result.

Is there a chance I mixed up the components? Let me check the original vectors again.

First vector: [-2, 0, 7]. Second vector: [3, 4, -5].

Yes, so components are:

-2, 0, 7 and 3, 4, -5.

So first components: -2 and 3. Second: 0 and 4. Third: 7 and -5. Correct.

No, I didn't mix up any components.

So all steps seem correct.

Alternatively, maybe I can think about distributive property or something? Let me try.

Suppose I consider the vectors as (-2, 0, 7) and (3, 4, -5).

Alternatively, maybe I can think of expanding some expression? Not sure.

Alternatively, if I make a mistake in sign? For example, if I thought that 7 * (-5) is positive 35? But I thought that's negative. Let me check again.

7 times negative 5: 7*5=35, and since one number is negative, the product is negative. So yes, -35.

So that term is definitely negative.

So adding -6, 0, and -35 gives me -41.

Therefore, I think my answer is correct.

I don't see any errors in my reasoning. So the dot product should be -41.

**Final Answer**
The dot product is \boxed{-41}.

To find the **dot product** of two vectors, we use the standard formula for the dot product in three-dimensional space:

$$
\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = a_1b_1 + a_2b_2 + a_3b_3
$$

---

### Step-by-step Calculation

We are given the vectors:

$$
\mathbf{a} = \begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}
$$

Now compute the dot product term by term:

1. **First components:**
   $ (-2) \times 3 = -6 $

2. **Second components:**
   $ 0 \times 4 = 0 $

3. **Third components:**
   $ 7 \times (-5) = -35 $

Now, sum the results:

$$
-6 + 0 + (-35) = -41
$$

---

### Final Answer

$$
\boxed{-41}
$$

Solution:
The dot product of $\begin{pmatrix} -2 \\ 0 \\ 7 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 4 \\ -5 \end{pmatrix}$ is
\[(-2)(3) + (0)(4) + (7)(-5) = \boxed{-41}.\]


Explanation:
Okay, so I need to find the difference between the positive square root of 64 and the cube root of 64. Hmm, let me think. First, let me make sure I understand the question correctly. It says "the positive square root of 64" and "the cube root of 64," and then subtract them? Wait, actually, it says "the difference between the positive square root of 64 and the cube root of 64." So, difference usually means subtracting one from the other. But I need to check which one is first. The wording is "difference between A and B," which is usually A - B. But sometimes people might interpret it differently. But I think in math, it's standard to take the first mentioned minus the second. Let me just confirm that later.

First, let me compute the positive square root of 64. The square root of a number is a value that, when multiplied by itself, gives the original number. So, sqrt(64). Well, 8 times 8 is 64, and also (-8) times (-8) is 64, but since they specified the positive square root, it must be 8. So, sqrt(64) = 8.

Next, the cube root of 64. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, cube root of 64. Let me think. 4 times 4 times 4 is 64? Wait, 4*4=16, 16*4=64. Yes, so 4 cubed is 64. Therefore, the cube root of 64 is 4. But wait, cube roots can also be negative, right? Because (-4)*(-4)*(-4) is -64. So, the cube root of 64 is 4, and the cube root of -64 is -4. But since the question just says "the cube root of 64," and doesn't specify, but in most cases, unless specified otherwise, the principal root is considered. For cube roots, the principal root is the real root, so it's 4. So, cube root of 64 is 4.

So, now the problem is asking for the difference between the positive square root of 64 (which is 8) and the cube root of 64 (which is 4). So, difference between them. If it's 8 minus 4, that would be 4. But wait, let me check if I got the order right. The question says "the difference between the positive square root of 64 and the cube root of 64." So, difference between A and B is A - B. So, 8 - 4 = 4. But maybe someone could interpret it as absolute difference? But in math, difference can sometimes mean the absolute value, but in algebra, when they say "difference between A and B," it's often A - B. However, if they wanted the absolute value, they might have said "the absolute difference." But since they mentioned "positive square root," which is already a positive number, and cube root of 64 is also positive, so the difference would be positive either way. So, 8 - 4 is 4, and 4 - 8 would be -4, but since they specified "positive square root," maybe they want 8 - 4.

But to make sure, let me check the exact wording again: "the difference between the positive square root of 64 and the cube root of 64." So, difference between A and B is A - B. So, positive square root of 64 is A, cube root of 64 is B. So, A - B. Therefore, 8 - 4 = 4. So, the answer is 4. But let me just verify once again.

Alternatively, perhaps the question is trying to trick me? Let me check if there's another interpretation. Maybe someone might think that cube root of 64 is something else? Wait, no. 4 cubed is 64. Let me check with exponents. 64 is 2^6. So, square root is 2^(6/2)=2^3=8. Cube root is 2^(6/3)=2^2=4. Yep, that checks out. So, square root is 8, cube root is 4. So, difference between them is 8 - 4 = 4. So, the answer is 4.

But let me think again. Wait, is there a possibility that someone might confuse the square root with the principal root? For example, sometimes square roots can have two values, positive and negative, but since they specified the positive square root, that's 8. The cube root of 64 is 4, as we said. So, 8 - 4 is 4. Therefore, the difference is 4.

Alternatively, maybe the question is written in a way that the difference is the absolute value? If so, it would still be 4. Either way, the answer is 4. So, I think that's the answer. Wait, but let me check with numbers again. If someone says difference between 8 and 4, it's 4. If they had said difference between 4 and 8, it's still 4. So, regardless of order, the absolute difference is 4. But since the question says "difference between A and B," where A is the positive square root (so 8) and B is the cube root (so 4), then mathematically difference is A - B = 8 - 4 = 4. So, that's the answer.

I think that's solid. So, the answer is 4. Therefore, the final answer is \boxed{4}.

**Final Answer**
The difference between the positive square root of 64 and the cube root of 64 is \boxed{4}.

To solve the problem, we evaluate two key roots and then compute their difference:

---

### Step 1: Positive Square Root of 64

The **positive square root** of a number is the non-negative number that, when squared, gives the original number.

$$
\sqrt{64} = 8
$$

This is because $8 \times 8 = 64$, and 8 is the **positive** root, as requested.

---

### Step 2: Cube Root of 64

The **cube root** of a number is a value that, when raised to the power of 3, gives the original number.

$$
\sqrt[3]{64} = 4
$$

This is because $4 \times 4 \times 4 = 64$. For cube roots, the principal root (in the real number system) is always the real value, which is positive in this case.

---

### Step 3: Compute the Difference

The question asks for the **difference between** the positive square root of 64 **and** the cube root of 64.

This is interpreted as:

$$
\text{Difference} = \text{Positive Square Root of } 64 - \text{Cube Root of } 64
$$

$$
\text{Difference} = 8 - 4 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the difference between the positive square root of 64 and the cube root of 64. Hmm, let me think. First, let me make sure I understand the question correctly. It says "the positive square root of 64" and "the cube root of 64," and then subtract them? Wait, actually, it says "the difference between the positive square root of 64 and the cube root of 64." So, difference usually means subtracting one from the other. But I need to check which one is first. The wording is "difference between A and B," which is usually A - B. But sometimes people might interpret it differently. But I think in math, it's standard to take the first mentioned minus the second. Let me just confirm that later.

First, let me compute the positive square root of 64. The square root of a number is a value that, when multiplied by itself, gives the original number. So, sqrt(64). Well, 8 times 8 is 64, and also (-8) times (-8) is 64, but since they specified the positive square root, it must be 8. So, sqrt(64) = 8.

Next, the cube root of 64. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, cube root of 64. Let me think. 4 times 4 times 4 is 64? Wait, 4*4=16, 16*4=64. Yes, so 4 cubed is 64. Therefore, the cube root of 64 is 4. But wait, cube roots can also be negative, right? Because (-4)*(-4)*(-4) is -64. So, the cube root of 64 is 4, and the cube root of -64 is -4. But since the question just says "the cube root of 64," and doesn't specify, but in most cases, unless specified otherwise, the principal root is considered. For cube roots, the principal root is the real root, so it's 4. So, cube root of 64 is 4.

So, now the problem is asking for the difference between the positive square root of 64 (which is 8) and the cube root of 64 (which is 4). So, difference between them. If it's 8 minus 4, that would be 4. But wait, let me check if I got the order right. The question says "the difference between the positive square root of 64 and the cube root of 64." So, difference between A and B is A - B. So, 8 - 4 = 4. But maybe someone could interpret it as absolute difference? But in math, difference can sometimes mean the absolute value, but in algebra, when they say "difference between A and B," it's often A - B. However, if they wanted the absolute value, they might have said "the absolute difference." But since they mentioned "positive square root," which is already a positive number, and cube root of 64 is also positive, so the difference would be positive either way. So, 8 - 4 is 4, and 4 - 8 would be -4, but since they specified "positive square root," maybe they want 8 - 4.

But to make sure, let me check the exact wording again: "the difference between the positive square root of 64 and the cube root of 64." So, difference between A and B is A - B. So, positive square root of 64 is A, cube root of 64 is B. So, A - B. Therefore, 8 - 4 = 4. So, the answer is 4. But let me just verify once again.

Alternatively, perhaps the question is trying to trick me? Let me check if there's another interpretation. Maybe someone might think that cube root of 64 is something else? Wait, no. 4 cubed is 64. Let me check with exponents. 64 is 2^6. So, square root is 2^(6/2)=2^3=8. Cube root is 2^(6/3)=2^2=4. Yep, that checks out. So, square root is 8, cube root is 4. So, difference between them is 8 - 4 = 4. So, the answer is 4.

But let me think again. Wait, is there a possibility that someone might confuse the square root with the principal root? For example, sometimes square roots can have two values, positive and negative, but since they specified the positive square root, that's 8. The cube root of 64 is 4, as we said. So, 8 - 4 is 4. Therefore, the difference is 4.

Alternatively, maybe the question is written in a way that the difference is the absolute value? If so, it would still be 4. Either way, the answer is 4. So, I think that's the answer. Wait, but let me check with numbers again. If someone says difference between 8 and 4, it's 4. If they had said difference between 4 and 8, it's still 4. So, regardless of order, the absolute difference is 4. But since the question says "difference between A and B," where A is the positive square root (so 8) and B is the cube root (so 4), then mathematically difference is A - B = 8 - 4 = 4. So, that's the answer.

I think that's solid. So, the answer is 4. Therefore, the final answer is \boxed{4}.

**Final Answer**
The difference between the positive square root of 64 and the cube root of 64 is \boxed{4}.

To solve the problem, we evaluate two key roots and then compute their difference:

---

### Step 1: Positive Square Root of 64

The **positive square root** of a number is the non-negative number that, when squared, gives the original number.

$$
\sqrt{64} = 8
$$

This is because $8 \times 8 = 64$, and 8 is the **positive** root, as requested.

---

### Step 2: Cube Root of 64

The **cube root** of a number is a value that, when raised to the power of 3, gives the original number.

$$
\sqrt[3]{64} = 4
$$

This is because $4 \times 4 \times 4 = 64$. For cube roots, the principal root (in the real number system) is always the real value, which is positive in this case.

---

### Step 3: Compute the Difference

The question asks for the **difference between** the positive square root of 64 **and** the cube root of 64.

This is interpreted as:

$$
\text{Difference} = \text{Positive Square Root of } 64 - \text{Cube Root of } 64
$$

$$
\text{Difference} = 8 - 4 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: The positive square root of 64 is $\sqrt{64}=8$. The cube root of 64 is $\sqrt[3]{64}=4$. The difference is $8-4=\boxed{4}$.


Explanation:
Okay, so I need to find the value of f(-2) for the function f(x) = 5x² + 3x + 4. Alright, let me think. First, I remember that when you have a function like this, to find f(-2), you just substitute x with -2 in the equation. Let me make sure I'm doing that right.

So the function is 5x squared plus 3x plus 4. So replacing x with -2 would mean plugging in -2 wherever there's an x. Let me write that out step by step.

First, let's rewrite the function with the substitution:

f(-2) = 5(-2)² + 3(-2) + 4

Okay, now I need to compute each term separately. Let me handle each part one by one to avoid mistakes.

Starting with the first term: 5(-2)². Wait, exponents come before multiplication in the order of operations, right? So I should calculate (-2) squared first.

Calculating (-2) squared: (-2) * (-2). Hmm, multiplying two negative numbers gives a positive result. So that would be 4. Then multiply that by 5. So 5 * 4. Let me do that: 5 * 4 is 20. So the first term is 20.

Alright, moving on to the second term: 3(-2). That's straightforward multiplication. 3 times -2. Well, positive times negative is negative, so that's -6.

Third term is just 4, so that stays as it is.

Now, putting all the terms together: 20 (from the first term) plus (-6) (from the second term) plus 4 (from the third term). Let me compute that step by step.

First, 20 + (-6) is the same as 20 - 6, which equals 14. Then, adding the last term: 14 + 4. That gives me 18.

Wait, let me check that again. Because sometimes I might mix up the signs. Let me verify each calculation again to be sure.

First term: (-2) squared is indeed 4. 5 multiplied by 4 is 20. Correct.

Second term: 3 multiplied by -2 is -6. Right.

Third term is 4, so adding them up: 20 - 6 + 4.

20 minus 6 is 14. Then 14 plus 4 is 18. Hmm, that seems correct.

But wait, let me think again. Maybe I should do the entire calculation in another way to cross-verify. Let's compute all terms again:

First term: 5*(-2)^2. The exponent is applied first, so (-2)^2 is 4, times 5 is 20.

Second term: 3*(-2) is -6.

Third term: +4.

So 20 - 6 + 4.

Alternatively, combining the constants: -6 + 4 is -2, then 20 + (-2) is 18. Yep, same result.

Hmm, seems like 18 is the answer. But just to be thorough, maybe I can plug it into another method or check my steps once more.

Alternatively, maybe expand the function step by step:

f(x) = 5x² + 3x + 4

So f(-2) = 5*(-2)^2 + 3*(-2) + 4

Calculating each term:

(-2)^2 = 4, so 5*4 = 20

3*(-2) = -6

Then add all terms: 20 + (-6) + 4

20 - 6 is 14, then 14 + 4 is 18.

Same answer.

Is there a possibility I messed up the sign somewhere? For example, if I had forgotten that (-2)^2 is positive? Let me check that.

Negative times negative is positive. So (-2) multiplied by (-2) is +4. So that term is definitely positive.

So 5 times 4 is 20. Then 3 times -2 is -6. Then +4.

So adding them up: 20 - 6 + 4. 20 - 6 is 14. 14 + 4 is 18.

Alternatively, maybe I can think of it as 20 + (-6 + 4). So -6 + 4 is -2, and 20 - 2 is 18. Yep.

I think that's solid.

Wait, just to check with another approach. Let me compute the entire expression as it is.

Original expression:

5(-2)^2 + 3(-2) + 4

Compute exponents first:

5*(4) + (-6) + 4

Multiply:

20 - 6 + 4

Then addition and subtraction left to right:

20 - 6 = 14

14 + 4 = 18

Same result.

I can't seem to find any mistake here. Maybe I can plug the value into a different form of the function? For example, expanding or factoring? Wait, the function is already in standard form, so maybe not. Alternatively, maybe using a different order of operations?

Alternatively, could I have misapplied the exponent? Let me think. If it was -2 squared, is there a different interpretation? Wait, sometimes people confuse -2 squared versus (-2) squared. But in this case, since it's written as (-2)^2, it's definitely (-2) squared, which is 4. If it was -2^2, that would be - (2^2) = -4, but that's not the case here.

So in our function, it's 5x², so when x is -2, it's 5*(-2)^2, which is 5*(4) = 20.

Therefore, all steps are correct.

Alternatively, maybe I can think of the function f(x) as a quadratic and plug in x=-2.

Another way: Maybe calculate using decimal or fractions? Not necessary here.

Alternatively, maybe using a calculator? But since I'm doing this manually, I need to trust my calculations.

Alternatively, maybe check with a different value. For instance, if I compute f(0), it should be 5*(0)^2 + 3*0 + 4 = 0 + 0 + 4 = 4. Which makes sense. If I compute f(1), it's 5(1) + 3(1) + 4 = 5 + 3 + 4 = 12.

So if I compute f(-1), that would be 5(-1)^2 + 3(-1) + 4 = 5(1) - 3 + 4 = 5 - 3 + 4 = 6.

So that's 6. Then f(-2) is 18? Let me see if there's a pattern or if that seems reasonable.

Looking at the function, as x becomes more negative, the x² term dominates, so the function should increase as x moves away from zero in the negative direction. So from x=-1 to x=-2, moving further left, the value increases from 6 to 18. That seems plausible.

Alternatively, if I compute f(-3), it would be 5*(-3)^2 + 3*(-3) + 4 = 5*9 - 9 + 4 = 45 - 9 + 4 = 40. Which is higher, so going from -1 (6) to -2 (18) to -3 (40), that seems consistent.

So that gives me more confidence that f(-2) is indeed 18.

Wait, but just to check, maybe I can use another method. For example, using the formula for quadratic functions. But I don't think that's necessary here.

Alternatively, maybe using the concept of function evaluation.

Alternatively, maybe think of the function as a polynomial and substitute.

But all methods lead me to 18. Therefore, I think the answer is 18.

But just to recap all steps once again:

Original function: f(x) = 5x² + 3x + 4

Substitute x = -2:

f(-2) = 5(-2)² + 3(-2) + 4

Calculate (-2)²: (-2) * (-2) = 4

Multiply by 5: 5 * 4 = 20

Calculate 3 * (-2): 3 * (-2) = -6

Add the constants: 20 + (-6) + 4

Compute 20 - 6 + 4:

First, 20 - 6 = 14

Then, 14 + 4 = 18

Yes, that's exactly what I did before.

I think I've verified it multiple times. Each time, I get 18. So unless I have a fundamental misunderstanding of how functions work, which I don't think I do, this should be correct.

Alternatively, let me check if the original problem was written correctly. The user said f(x) = 5x² + 3x + 4. So no, there's no typo here.

Therefore, I think confidently that f(-2) equals 18.

**Final Answer**
The value of $ f(-2) $ is \boxed{18}.

To evaluate the function $ f(x) = 5x^2 + 3x + 4 $ at $ x = -2 $, we substitute $ x $ with $ -2 $ in the expression. This involves a careful application of order of operations and handling of negative numbers.

---

### Step 1: Substitute $ x = -2 $ into the function

$$
f(-2) = 5(-2)^2 + 3(-2) + 4
$$

---

### Step 2: Evaluate each term

- **First term**: $ (-2)^2 = 4 $, so:
  $$
  5(-2)^2 = 5 \cdot 4 = 20
  $$

- **Second term**:
  $$
  3(-2) = -6
  $$

- **Third term** remains the same:
  $$
  +4
  $$

---

### Step 3: Combine all the terms

$$
f(-2) = 20 + (-6) + 4
$$

$$
f(-2) = 20 - 6 + 4
$$

$$
f(-2) = (20 - 6) + 4 = 14 + 4 = 18
$$

---

### Final Answer

$$
\boxed{18}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of f(-2) for the function f(x) = 5x² + 3x + 4. Alright, let me think. First, I remember that when you have a function like this, to find f(-2), you just substitute x with -2 in the equation. Let me make sure I'm doing that right.

So the function is 5x squared plus 3x plus 4. So replacing x with -2 would mean plugging in -2 wherever there's an x. Let me write that out step by step.

First, let's rewrite the function with the substitution:

f(-2) = 5(-2)² + 3(-2) + 4

Okay, now I need to compute each term separately. Let me handle each part one by one to avoid mistakes.

Starting with the first term: 5(-2)². Wait, exponents come before multiplication in the order of operations, right? So I should calculate (-2) squared first.

Calculating (-2) squared: (-2) * (-2). Hmm, multiplying two negative numbers gives a positive result. So that would be 4. Then multiply that by 5. So 5 * 4. Let me do that: 5 * 4 is 20. So the first term is 20.

Alright, moving on to the second term: 3(-2). That's straightforward multiplication. 3 times -2. Well, positive times negative is negative, so that's -6.

Third term is just 4, so that stays as it is.

Now, putting all the terms together: 20 (from the first term) plus (-6) (from the second term) plus 4 (from the third term). Let me compute that step by step.

First, 20 + (-6) is the same as 20 - 6, which equals 14. Then, adding the last term: 14 + 4. That gives me 18.

Wait, let me check that again. Because sometimes I might mix up the signs. Let me verify each calculation again to be sure.

First term: (-2) squared is indeed 4. 5 multiplied by 4 is 20. Correct.

Second term: 3 multiplied by -2 is -6. Right.

Third term is 4, so adding them up: 20 - 6 + 4.

20 minus 6 is 14. Then 14 plus 4 is 18. Hmm, that seems correct.

But wait, let me think again. Maybe I should do the entire calculation in another way to cross-verify. Let's compute all terms again:

First term: 5*(-2)^2. The exponent is applied first, so (-2)^2 is 4, times 5 is 20.

Second term: 3*(-2) is -6.

Third term: +4.

So 20 - 6 + 4.

Alternatively, combining the constants: -6 + 4 is -2, then 20 + (-2) is 18. Yep, same result.

Hmm, seems like 18 is the answer. But just to be thorough, maybe I can plug it into another method or check my steps once more.

Alternatively, maybe expand the function step by step:

f(x) = 5x² + 3x + 4

So f(-2) = 5*(-2)^2 + 3*(-2) + 4

Calculating each term:

(-2)^2 = 4, so 5*4 = 20

3*(-2) = -6

Then add all terms: 20 + (-6) + 4

20 - 6 is 14, then 14 + 4 is 18.

Same answer.

Is there a possibility I messed up the sign somewhere? For example, if I had forgotten that (-2)^2 is positive? Let me check that.

Negative times negative is positive. So (-2) multiplied by (-2) is +4. So that term is definitely positive.

So 5 times 4 is 20. Then 3 times -2 is -6. Then +4.

So adding them up: 20 - 6 + 4. 20 - 6 is 14. 14 + 4 is 18.

Alternatively, maybe I can think of it as 20 + (-6 + 4). So -6 + 4 is -2, and 20 - 2 is 18. Yep.

I think that's solid.

Wait, just to check with another approach. Let me compute the entire expression as it is.

Original expression:

5(-2)^2 + 3(-2) + 4

Compute exponents first:

5*(4) + (-6) + 4

Multiply:

20 - 6 + 4

Then addition and subtraction left to right:

20 - 6 = 14

14 + 4 = 18

Same result.

I can't seem to find any mistake here. Maybe I can plug the value into a different form of the function? For example, expanding or factoring? Wait, the function is already in standard form, so maybe not. Alternatively, maybe using a different order of operations?

Alternatively, could I have misapplied the exponent? Let me think. If it was -2 squared, is there a different interpretation? Wait, sometimes people confuse -2 squared versus (-2) squared. But in this case, since it's written as (-2)^2, it's definitely (-2) squared, which is 4. If it was -2^2, that would be - (2^2) = -4, but that's not the case here.

So in our function, it's 5x², so when x is -2, it's 5*(-2)^2, which is 5*(4) = 20.

Therefore, all steps are correct.

Alternatively, maybe I can think of the function f(x) as a quadratic and plug in x=-2.

Another way: Maybe calculate using decimal or fractions? Not necessary here.

Alternatively, maybe using a calculator? But since I'm doing this manually, I need to trust my calculations.

Alternatively, maybe check with a different value. For instance, if I compute f(0), it should be 5*(0)^2 + 3*0 + 4 = 0 + 0 + 4 = 4. Which makes sense. If I compute f(1), it's 5(1) + 3(1) + 4 = 5 + 3 + 4 = 12.

So if I compute f(-1), that would be 5(-1)^2 + 3(-1) + 4 = 5(1) - 3 + 4 = 5 - 3 + 4 = 6.

So that's 6. Then f(-2) is 18? Let me see if there's a pattern or if that seems reasonable.

Looking at the function, as x becomes more negative, the x² term dominates, so the function should increase as x moves away from zero in the negative direction. So from x=-1 to x=-2, moving further left, the value increases from 6 to 18. That seems plausible.

Alternatively, if I compute f(-3), it would be 5*(-3)^2 + 3*(-3) + 4 = 5*9 - 9 + 4 = 45 - 9 + 4 = 40. Which is higher, so going from -1 (6) to -2 (18) to -3 (40), that seems consistent.

So that gives me more confidence that f(-2) is indeed 18.

Wait, but just to check, maybe I can use another method. For example, using the formula for quadratic functions. But I don't think that's necessary here.

Alternatively, maybe using the concept of function evaluation.

Alternatively, maybe think of the function as a polynomial and substitute.

But all methods lead me to 18. Therefore, I think the answer is 18.

But just to recap all steps once again:

Original function: f(x) = 5x² + 3x + 4

Substitute x = -2:

f(-2) = 5(-2)² + 3(-2) + 4

Calculate (-2)²: (-2) * (-2) = 4

Multiply by 5: 5 * 4 = 20

Calculate 3 * (-2): 3 * (-2) = -6

Add the constants: 20 + (-6) + 4

Compute 20 - 6 + 4:

First, 20 - 6 = 14

Then, 14 + 4 = 18

Yes, that's exactly what I did before.

I think I've verified it multiple times. Each time, I get 18. So unless I have a fundamental misunderstanding of how functions work, which I don't think I do, this should be correct.

Alternatively, let me check if the original problem was written correctly. The user said f(x) = 5x² + 3x + 4. So no, there's no typo here.

Therefore, I think confidently that f(-2) equals 18.

**Final Answer**
The value of $ f(-2) $ is \boxed{18}.

To evaluate the function $ f(x) = 5x^2 + 3x + 4 $ at $ x = -2 $, we substitute $ x $ with $ -2 $ in the expression. This involves a careful application of order of operations and handling of negative numbers.

---

### Step 1: Substitute $ x = -2 $ into the function

$$
f(-2) = 5(-2)^2 + 3(-2) + 4
$$

---

### Step 2: Evaluate each term

- **First term**: $ (-2)^2 = 4 $, so:
  $$
  5(-2)^2 = 5 \cdot 4 = 20
  $$

- **Second term**:
  $$
  3(-2) = -6
  $$

- **Third term** remains the same:
  $$
  +4
  $$

---

### Step 3: Combine all the terms

$$
f(-2) = 20 + (-6) + 4
$$

$$
f(-2) = 20 - 6 + 4
$$

$$
f(-2) = (20 - 6) + 4 = 14 + 4 = 18
$$

---

### Final Answer

$$
\boxed{18}
$$

Solution: We have $f(-2) = 5(-2)^2 + 3(-2) + 4 = 5(4) -6 + 4 = \boxed{18}$.