Explanation:
Okay, so I need to find the sum of all multiples of 7 between 100 and 200. Hmm, let me think about how to approach this. First, I remember that multiples of 7 are numbers like 7, 14, 21, and so on. But here, we need those multiples that are specifically between 100 and 200. So, I need to figure out which numbers in that range are divisible by 7 and then add them all up.

Let me start by finding the smallest multiple of 7 that is greater than or equal to 100. How do I do that? Well, if I divide 100 by 7, that should give me a number which I can use to find the next multiple. Let me calculate 100 divided by 7. Let me do that division. 7 times 14 is 98, right? Because 7*10 is 70, 7*14 is 70 + 28 = 98. So, 100 divided by 7 is 14 with a remainder of 2. So, that means the next multiple of 7 after 98 is 98 + 7 = 105. Therefore, 105 is the first multiple of 7 that is greater than or equal to 100.

Now, I need to check if 105 is less than 200. Well, 105 is definitely less than 200, so that's good. Now, I need to find the last multiple of 7 that is less than or equal to 200. Let me do 200 divided by 7. Let me compute that. 7*28 is 196, because 7*25 is 175, and 7*3 is 21, so 175 + 21 is 196. Then 200 - 196 is 4. So, the remainder is 4. Therefore, the last multiple of 7 before 200 is 196. So, the multiples of 7 between 100 and 200 are from 105 up to 196, inclusive.

Now, I need to figure out how many terms there are in this sequence. To do that, maybe I can use the formula for the nth term of an arithmetic sequence. The formula is a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the number of terms.

In this case, the first term a_1 is 105, the common difference d is 7, and the last term a_n is 196. Let me plug these into the formula and solve for n.

So, 196 = 105 + (n - 1)*7.

Let me subtract 105 from both sides:

196 - 105 = (n - 1)*7.

Calculating 196 - 105: 196 - 100 is 96, minus 5 more is 91. So, 91 = (n - 1)*7.

Divide both sides by 7: 91 / 7 = n - 1. 91 divided by 7 is 13. So, 13 = n - 1. Therefore, n = 14. Wait, so there are 14 terms? Let me check that.

Alternatively, maybe I can calculate the number of terms by taking the difference between the first and last term, dividing by the common difference, and then adding 1. Let me try that.

So, the formula for the number of terms is ((last term - first term)/common difference) + 1. So, ((196 - 105)/7) + 1. Let's compute 196 - 105. That's 91. Then 91 / 7 is 13. Then adding 1 gives 14. Yep, same result. So there are 14 terms. Okay, that seems right.

Now, once I know how many terms there are, I can use the formula for the sum of an arithmetic sequence. The formula is S_n = n/2 * (a_1 + a_n). So, the sum is equal to the number of terms divided by 2 times the sum of the first and last term.

Plugging in the numbers: S_n = 14/2 * (105 + 196). That simplifies to 7 * (301). Let me calculate 7 * 301. Hmm, 7 * 300 is 2100, and 7 * 1 is 7, so 2100 + 7 is 2107. Therefore, the sum should be 2107.

Wait a second, let me verify that. Because sometimes when I do arithmetic, I might make a mistake. Let me recalculate 105 + 196. 105 + 196: 100 + 196 is 296, plus 5 is 301. Then 7 * 301. Let me do 300*7 + 1*7 = 2100 + 7 = 2107. Yeah, that's correct.

Alternatively, maybe I can check by listing all the terms and adding them up? Although that might take longer, but just to be sure. Let's try that.

Starting with 105, then adding 7 each time:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me count them: 1. 105, 2. 112, 3. 119, 4. 126, 5. 133, 6. 140, 7. 147, 8. 154, 9. 161, 10. 168, 11. 175, 12. 182, 13. 189, 14. 196. Yep, 14 terms. Now, let me add them up step by step.

Starting with 105 + 112 = 217

217 + 119 = 336

336 + 126 = 462

462 + 133 = 595

595 + 140 = 735

735 + 147 = 882

882 + 154 = 1036

1036 + 161 = 1197

1197 + 168 = 1365

1365 + 175 = 1540

1540 + 182 = 1722

1722 + 189 = 1911

1911 + 196 = 2107

Yes! So, the total is 2107. That matches the earlier calculation. So that seems correct.

But just to make sure I didn't make any mistakes in my initial steps, let me go through them again.

First, finding the first multiple of 7 above 100: 100 divided by 7 is approximately 14.2857. So the next integer is 15, so 15*7 = 105. Correct. Then the last multiple of 7 below 200: 200 divided by 7 is approximately 28.571, so the integer part is 28, so 28*7 = 196. Correct.

Then, the number of terms: ((196 - 105)/7) + 1. That's (91/7)+1 =13+1=14. Correct. Then the sum is 14/2*(105+196)=7*(301)=2107. Correct. And when I added all the terms step by step, I got 2107. So, seems solid.

Alternatively, is there another way to approach this problem? Maybe using the formula for the sum of multiples? Let me think. If I consider the multiples of 7 between 1 and 200, and then subtract the multiples of 7 between 1 and 100. Wait, but the original question is between 100 and 200, inclusive? Wait, the question says "between 100 and 200". Wait, does "between" include 100 and 200 or not?

Wait, the problem says "multiples of 7 between 100 and 200". Depending on interpretation, sometimes "between" can be exclusive, but in math problems, sometimes it's inclusive. But in my earlier calculation, I included 105 and 196. But if "between" was exclusive, then it would be numbers strictly greater than 100 and strictly less than 200. However, the problem says "between 100 and 200", but since 100 and 200 are not multiples of 7, maybe it's okay. Let me check.

First, check if 100 is a multiple of 7: 7*14=98, 7*15=105. So, 100 is not a multiple of 7. Similarly, 200 divided by 7 is 28.571..., so not a multiple of 7. So whether we include them or not doesn't affect the result. Therefore, whether "between" is inclusive or exclusive, the answer remains the same. Because the endpoints aren't multiples of 7. Therefore, the initial calculation is correct.

Alternatively, if someone thought that between 100 and 200 includes 100 and 200, but since they aren't multiples of 7, it doesn't matter. So the answer is still 2107.

But just to double-check, let me see. Suppose someone thought that between 100 and 200 inclusive, then the multiples would still be 105 to 196, same as before. So, the answer remains same.

Alternatively, if someone thought that between 100 and 200 exclusive, meaning numbers from 101 to 199, but again, since 105 is the first multiple above 100, and 196 is the last multiple below 200, so same numbers. So, regardless of interpretation, the numbers are same. Therefore, correct.

Another way to think about it: The formula for the sum of an arithmetic series. So if I can find the first term, last term, and number of terms, then compute accordingly. Which I did.

Alternatively, think in terms of how many multiples of 7 there are between 1 and 200, subtract those between 1 and 100. Let's try that approach.

First, how many multiples of 7 are there between 1 and 200? Well, floor(200/7) = 28, since 28*7=196, which is less than 200. Similarly, multiples of 7 between 1 and 100: floor(100/7) =14, since 14*7=98, which is less than 100. Therefore, the number of multiples between 100 and 200 is 28 -14=14. Which matches our previous result. Then the sum of multiples of 7 up to 200 is 7*(1 + 2 + ... +28) =7*(28*29)/2. Wait, no. Wait, the sum of the first n multiples of 7 is 7*(1 + 2 + ... +n) =7*(n(n+1))/2. Similarly, the sum up to 100 is 7*(1 + 2 + ... +14)=7*(14*15)/2. Therefore, the sum between 100 and 200 is sum up to 200 minus sum up to 100. Let me compute that.

First, compute sum up to 200: n=28, so 7*(28*29)/2. Let me compute that. First compute 28*29: 28*30=840, minus 28=812. Then 7*(812)/2. Let me compute 812/2=406, then 406*7. 400*7=2800, 6*7=42, so total 2800+42=2842. So sum up to 200 is 2842.

Then sum up to 100: n=14, so 7*(14*15)/2. Let me calculate that. 14*15=210. Then 210/2=105. Then 105*7=735. Therefore, the sum between 100 and 200 is 2842 - 735 = ?

Compute 2842 - 700 = 2142, then subtract 35 more: 2142 -35=2107. Yep, same result. So this method also gives 2107. So that confirms it again.

Therefore, regardless of the method, the answer is 2107. Therefore, I can be confident that this is correct.

But let me just think if there is another possible way that someone might approach this problem incorrectly, maybe leading to a different answer? For example, if someone thought that the first multiple is 100, but 100 isn't a multiple of 7. If someone tried to adjust, but I don't think so. Or if someone miscalculates the number of terms. For example, if someone thought that the number of terms is 14, but if they did ((last - first)/difference) without adding 1, which would be 13, then the sum would be 13/2*(105 + 196) which is 6.5*301=1956.5, which is not possible, since the sum of integers must be integer. So that would be wrong. But since we added 1, we got the correct number of terms.

Alternatively, if someone messed up the first term or last term. For example, if they thought the first multiple was 105 or 112? But no, since 105 is the first multiple above 100. Similarly, if someone thought 196 is not included? But since 196 is less than 200, it is included.

Alternatively, maybe miscalculating the sum. For example, if someone thought that (a1 + an) is 105 + 196 = 301, then multiplied by n=14, divided by 2. Which is 14/2=7, 7*301=2107. Correct. So that's right.

Alternatively, if someone thought that the common difference is 1 instead of 7? But no, the difference between multiples of 7 is 7. So that would be a mistake, but not in this case.

Therefore, all checks lead to 2107. Therefore, I think that's the correct answer.

**Final Answer**
The sum of all multiples of 7 between 100 and 200 is \boxed{2107}.

To solve the problem of finding the sum of all **multiples of 7 between 100 and 200**, we approach it step by step using the properties of arithmetic sequences.

---

### Step 1: Identify the First and Last Multiples of 7 in the Range

We are looking for all **multiples of 7** such that $100 < \text{multiple of } 7 < 200$. Since 100 and 200 are not multiples of 7, we need to find the smallest and largest such numbers:

- **First multiple of 7 ≥ 100:**
  $$
  \frac{100}{7} = 14.2857\ldots \Rightarrow \text{Next integer } = 15 \Rightarrow 15 \times 7 = 105
  $$

- **Last multiple of 7 ≤ 200:**
  $$
  \frac{200}{7} = 28.571\ldots \Rightarrow \text{Floor value } = 28 \Rightarrow 28 \times 7 = 196
  $$

So, the sequence of multiples of 7 between 100 and 200 is:

$$
105, 112, 119, \ldots, 196
$$

---

### Step 2: Determine the Number of Terms in the Sequence

This is an arithmetic sequence with:

- First term $a = 105$
- Last term $l = 196$
- Common difference $d = 7$

We use the formula to find the number of terms $n$:
$$
l = a + (n - 1)d \\
196 = 105 + (n - 1) \cdot 7 \\
196 - 105 = (n - 1) \cdot 7 \\
91 = (n - 1) \cdot 7 \\
n - 1 = \frac{91}{7} = 13 \Rightarrow n = 14
$$

So, there are **14 terms** in the sequence.

---

### Step 3: Calculate the Sum of the Sequence

We use the formula for the sum of an arithmetic sequence:

$$
S_n = \frac{n}{2} \cdot (a + l)
$$

Substitute known values:

$$
S = \frac{14}{2} \cdot (105 + 196) = 7 \cdot 301 = 2107
$$

---

### Step 4: Confirm the Result

- **Manual verification** by adding the numbers in the sequence (105 through 196 by steps of 7) yields the same result: 2107.
- **Alternative method**: Compute the sum of the first 28 multiples of 7 and subtract the sum of the first 14 multiples of 7 also gives 2107.

Each step confirms the total.

---

### Final Answer

$$
\boxed{2107}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of all multiples of 7 between 100 and 200. Hmm, let me think about how to approach this. First, I remember that multiples of 7 are numbers like 7, 14, 21, and so on. But here, we need those multiples that are specifically between 100 and 200. So, I need to figure out which numbers in that range are divisible by 7 and then add them all up.

Let me start by finding the smallest multiple of 7 that is greater than or equal to 100. How do I do that? Well, if I divide 100 by 7, that should give me a number which I can use to find the next multiple. Let me calculate 100 divided by 7. Let me do that division. 7 times 14 is 98, right? Because 7*10 is 70, 7*14 is 70 + 28 = 98. So, 100 divided by 7 is 14 with a remainder of 2. So, that means the next multiple of 7 after 98 is 98 + 7 = 105. Therefore, 105 is the first multiple of 7 that is greater than or equal to 100.

Now, I need to check if 105 is less than 200. Well, 105 is definitely less than 200, so that's good. Now, I need to find the last multiple of 7 that is less than or equal to 200. Let me do 200 divided by 7. Let me compute that. 7*28 is 196, because 7*25 is 175, and 7*3 is 21, so 175 + 21 is 196. Then 200 - 196 is 4. So, the remainder is 4. Therefore, the last multiple of 7 before 200 is 196. So, the multiples of 7 between 100 and 200 are from 105 up to 196, inclusive.

Now, I need to figure out how many terms there are in this sequence. To do that, maybe I can use the formula for the nth term of an arithmetic sequence. The formula is a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the number of terms.

In this case, the first term a_1 is 105, the common difference d is 7, and the last term a_n is 196. Let me plug these into the formula and solve for n.

So, 196 = 105 + (n - 1)*7.

Let me subtract 105 from both sides:

196 - 105 = (n - 1)*7.

Calculating 196 - 105: 196 - 100 is 96, minus 5 more is 91. So, 91 = (n - 1)*7.

Divide both sides by 7: 91 / 7 = n - 1. 91 divided by 7 is 13. So, 13 = n - 1. Therefore, n = 14. Wait, so there are 14 terms? Let me check that.

Alternatively, maybe I can calculate the number of terms by taking the difference between the first and last term, dividing by the common difference, and then adding 1. Let me try that.

So, the formula for the number of terms is ((last term - first term)/common difference) + 1. So, ((196 - 105)/7) + 1. Let's compute 196 - 105. That's 91. Then 91 / 7 is 13. Then adding 1 gives 14. Yep, same result. So there are 14 terms. Okay, that seems right.

Now, once I know how many terms there are, I can use the formula for the sum of an arithmetic sequence. The formula is S_n = n/2 * (a_1 + a_n). So, the sum is equal to the number of terms divided by 2 times the sum of the first and last term.

Plugging in the numbers: S_n = 14/2 * (105 + 196). That simplifies to 7 * (301). Let me calculate 7 * 301. Hmm, 7 * 300 is 2100, and 7 * 1 is 7, so 2100 + 7 is 2107. Therefore, the sum should be 2107.

Wait a second, let me verify that. Because sometimes when I do arithmetic, I might make a mistake. Let me recalculate 105 + 196. 105 + 196: 100 + 196 is 296, plus 5 is 301. Then 7 * 301. Let me do 300*7 + 1*7 = 2100 + 7 = 2107. Yeah, that's correct.

Alternatively, maybe I can check by listing all the terms and adding them up? Although that might take longer, but just to be sure. Let's try that.

Starting with 105, then adding 7 each time:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me count them: 1. 105, 2. 112, 3. 119, 4. 126, 5. 133, 6. 140, 7. 147, 8. 154, 9. 161, 10. 168, 11. 175, 12. 182, 13. 189, 14. 196. Yep, 14 terms. Now, let me add them up step by step.

Starting with 105 + 112 = 217

217 + 119 = 336

336 + 126 = 462

462 + 133 = 595

595 + 140 = 735

735 + 147 = 882

882 + 154 = 1036

1036 + 161 = 1197

1197 + 168 = 1365

1365 + 175 = 1540

1540 + 182 = 1722

1722 + 189 = 1911

1911 + 196 = 2107

Yes! So, the total is 2107. That matches the earlier calculation. So that seems correct.

But just to make sure I didn't make any mistakes in my initial steps, let me go through them again.

First, finding the first multiple of 7 above 100: 100 divided by 7 is approximately 14.2857. So the next integer is 15, so 15*7 = 105. Correct. Then the last multiple of 7 below 200: 200 divided by 7 is approximately 28.571, so the integer part is 28, so 28*7 = 196. Correct.

Then, the number of terms: ((196 - 105)/7) + 1. That's (91/7)+1 =13+1=14. Correct. Then the sum is 14/2*(105+196)=7*(301)=2107. Correct. And when I added all the terms step by step, I got 2107. So, seems solid.

Alternatively, is there another way to approach this problem? Maybe using the formula for the sum of multiples? Let me think. If I consider the multiples of 7 between 1 and 200, and then subtract the multiples of 7 between 1 and 100. Wait, but the original question is between 100 and 200, inclusive? Wait, the question says "between 100 and 200". Wait, does "between" include 100 and 200 or not?

Wait, the problem says "multiples of 7 between 100 and 200". Depending on interpretation, sometimes "between" can be exclusive, but in math problems, sometimes it's inclusive. But in my earlier calculation, I included 105 and 196. But if "between" was exclusive, then it would be numbers strictly greater than 100 and strictly less than 200. However, the problem says "between 100 and 200", but since 100 and 200 are not multiples of 7, maybe it's okay. Let me check.

First, check if 100 is a multiple of 7: 7*14=98, 7*15=105. So, 100 is not a multiple of 7. Similarly, 200 divided by 7 is 28.571..., so not a multiple of 7. So whether we include them or not doesn't affect the result. Therefore, whether "between" is inclusive or exclusive, the answer remains the same. Because the endpoints aren't multiples of 7. Therefore, the initial calculation is correct.

Alternatively, if someone thought that between 100 and 200 includes 100 and 200, but since they aren't multiples of 7, it doesn't matter. So the answer is still 2107.

But just to double-check, let me see. Suppose someone thought that between 100 and 200 inclusive, then the multiples would still be 105 to 196, same as before. So, the answer remains same.

Alternatively, if someone thought that between 100 and 200 exclusive, meaning numbers from 101 to 199, but again, since 105 is the first multiple above 100, and 196 is the last multiple below 200, so same numbers. So, regardless of interpretation, the numbers are same. Therefore, correct.

Another way to think about it: The formula for the sum of an arithmetic series. So if I can find the first term, last term, and number of terms, then compute accordingly. Which I did.

Alternatively, think in terms of how many multiples of 7 there are between 1 and 200, subtract those between 1 and 100. Let's try that approach.

First, how many multiples of 7 are there between 1 and 200? Well, floor(200/7) = 28, since 28*7=196, which is less than 200. Similarly, multiples of 7 between 1 and 100: floor(100/7) =14, since 14*7=98, which is less than 100. Therefore, the number of multiples between 100 and 200 is 28 -14=14. Which matches our previous result. Then the sum of multiples of 7 up to 200 is 7*(1 + 2 + ... +28) =7*(28*29)/2. Wait, no. Wait, the sum of the first n multiples of 7 is 7*(1 + 2 + ... +n) =7*(n(n+1))/2. Similarly, the sum up to 100 is 7*(1 + 2 + ... +14)=7*(14*15)/2. Therefore, the sum between 100 and 200 is sum up to 200 minus sum up to 100. Let me compute that.

First, compute sum up to 200: n=28, so 7*(28*29)/2. Let me compute that. First compute 28*29: 28*30=840, minus 28=812. Then 7*(812)/2. Let me compute 812/2=406, then 406*7. 400*7=2800, 6*7=42, so total 2800+42=2842. So sum up to 200 is 2842.

Then sum up to 100: n=14, so 7*(14*15)/2. Let me calculate that. 14*15=210. Then 210/2=105. Then 105*7=735. Therefore, the sum between 100 and 200 is 2842 - 735 = ?

Compute 2842 - 700 = 2142, then subtract 35 more: 2142 -35=2107. Yep, same result. So this method also gives 2107. So that confirms it again.

Therefore, regardless of the method, the answer is 2107. Therefore, I can be confident that this is correct.

But let me just think if there is another possible way that someone might approach this problem incorrectly, maybe leading to a different answer? For example, if someone thought that the first multiple is 100, but 100 isn't a multiple of 7. If someone tried to adjust, but I don't think so. Or if someone miscalculates the number of terms. For example, if someone thought that the number of terms is 14, but if they did ((last - first)/difference) without adding 1, which would be 13, then the sum would be 13/2*(105 + 196) which is 6.5*301=1956.5, which is not possible, since the sum of integers must be integer. So that would be wrong. But since we added 1, we got the correct number of terms.

Alternatively, if someone messed up the first term or last term. For example, if they thought the first multiple was 105 or 112? But no, since 105 is the first multiple above 100. Similarly, if someone thought 196 is not included? But since 196 is less than 200, it is included.

Alternatively, maybe miscalculating the sum. For example, if someone thought that (a1 + an) is 105 + 196 = 301, then multiplied by n=14, divided by 2. Which is 14/2=7, 7*301=2107. Correct. So that's right.

Alternatively, if someone thought that the common difference is 1 instead of 7? But no, the difference between multiples of 7 is 7. So that would be a mistake, but not in this case.

Therefore, all checks lead to 2107. Therefore, I think that's the correct answer.

**Final Answer**
The sum of all multiples of 7 between 100 and 200 is \boxed{2107}.

To solve the problem of finding the sum of all **multiples of 7 between 100 and 200**, we approach it step by step using the properties of arithmetic sequences.

---

### Step 1: Identify the First and Last Multiples of 7 in the Range

We are looking for all **multiples of 7** such that $100 < \text{multiple of } 7 < 200$. Since 100 and 200 are not multiples of 7, we need to find the smallest and largest such numbers:

- **First multiple of 7 ≥ 100:**
  $$
  \frac{100}{7} = 14.2857\ldots \Rightarrow \text{Next integer } = 15 \Rightarrow 15 \times 7 = 105
  $$

- **Last multiple of 7 ≤ 200:**
  $$
  \frac{200}{7} = 28.571\ldots \Rightarrow \text{Floor value } = 28 \Rightarrow 28 \times 7 = 196
  $$

So, the sequence of multiples of 7 between 100 and 200 is:

$$
105, 112, 119, \ldots, 196
$$

---

### Step 2: Determine the Number of Terms in the Sequence

This is an arithmetic sequence with:

- First term $a = 105$
- Last term $l = 196$
- Common difference $d = 7$

We use the formula to find the number of terms $n$:
$$
l = a + (n - 1)d \\
196 = 105 + (n - 1) \cdot 7 \\
196 - 105 = (n - 1) \cdot 7 \\
91 = (n - 1) \cdot 7 \\
n - 1 = \frac{91}{7} = 13 \Rightarrow n = 14
$$

So, there are **14 terms** in the sequence.

---

### Step 3: Calculate the Sum of the Sequence

We use the formula for the sum of an arithmetic sequence:

$$
S_n = \frac{n}{2} \cdot (a + l)
$$

Substitute known values:

$$
S = \frac{14}{2} \cdot (105 + 196) = 7 \cdot 301 = 2107
$$

---

### Step 4: Confirm the Result

- **Manual verification** by adding the numbers in the sequence (105 through 196 by steps of 7) yields the same result: 2107.
- **Alternative method**: Compute the sum of the first 28 multiples of 7 and subtract the sum of the first 14 multiples of 7 also gives 2107.

Each step confirms the total.

---

### Final Answer

$$
\boxed{2107}
$$

Solution:
The smallest multiple of 7 between 100 and 200 is 105, and the largest multiple is 196.  Thus, we want to find the sum of the arithmetic series $105 + 112 + \dots + 196$.

The $n^{\text{th}}$ term in this arithmetic sequence is $105 + 7(n - 1) = 7n + 98$.  If $7n + 98 = 196$, then $n = 14$, so the number of terms in this sequence is 14.

The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum is $(105 + 196)/2 \cdot 14 = \boxed{2107}$.


Explanation:
Okay, so I need to rewrite the quadratic equation $x^2 + 2.6x + 3.6$ in the form $(x + b)^2 + c$, where $b$ and $c$ are constants. Then, I have to find $b + c$ as a decimal. Hmm, let me think about how to do this.

First, I remember that completing the square is the method used to convert a quadratic from standard form $ax^2 + bx + c$ to vertex form $(x + b)^2 + c$. Wait, actually, the vertex form is usually written as $a(x - h)^2 + k$, but in this case, the problem specifies $(x + b)^2 + c$, so maybe they just want it in a similar format, where the coefficient of $x^2$ is 1. Since the original quadratic has a leading coefficient of 1, that should work out.

So, let me recall the steps for completing the square. For a quadratic $x^2 + bx + c$, you take half of the coefficient of $x$, square it, and add and subtract that value to complete the square. Let me write that down.

Given $x^2 + 2.6x + 3.6$, the coefficient of $x$ is 2.6. So, half of 2.6 is... let me compute that. Half of 2.6 is 1.3. Then, squaring that gives me $1.3^2$. Let me calculate that. 1.3 times 1.3. Well, 1 times 1 is 1, 1 times 0.3 is 0.3, 0.3 times 1 is another 0.3, and 0.3 times 0.3 is 0.09. Adding those up: 1 + 0.3 + 0.3 + 0.09 = 1.69. So, $1.3^2 = 1.69$.

So, to complete the square, I need to add and subtract 1.69 inside the equation. Let me write that out:

Original expression: $x^2 + 2.6x + 3.6$

Let me rewrite it as $x^2 + 2.6x + 1.69 - 1.69 + 3.6$. Wait, but I need to make sure that adding and subtracting the same value keeps the equation equivalent. So:

$x^2 + 2.6x + 3.6 = (x^2 + 2.6x + 1.69) - 1.69 + 3.6$

Now, the first three terms form a perfect square trinomial. Let me check that. If I take $(x + 1.3)^2$, that would be $x^2 + 2*1.3x + (1.3)^2$, which is $x^2 + 2.6x + 1.69$. Yes! That's exactly what we have. So, the expression becomes:

$(x + 1.3)^2 - 1.69 + 3.6$

Now, let me compute $-1.69 + 3.6$. Let's subtract 1.69 from 3.6.

First, 3.6 minus 1.69. Let me write that as 3.60 - 1.69. Subtracting the decimals:

Starting with the hundredths place: 0 - 9. I can't do that, so I need to borrow. The tenths place is 6, so borrow 1 from the tenths place, making it 5, and the hundredths place becomes 10. Now, 10 - 9 = 1. Then, tenths place: 5 - 6. Again, can't do that, so borrow from the units place. The units place is 3, so borrow 1, making it 2, and the tenths place becomes 15. Then, 15 - 6 = 9. Finally, units place: 2 - 1 = 1. Wait, let me check again:

Wait, original numbers: 3.60 minus 1.69.

First, think of 3.60 as 3.6 and 1.69 as 1.69. So, 3.60 - 1.69. Let's subtract 1.69 from 3.60.

Alternatively, think of it as 3.60 - 1.69 = (3.60 - 1.60) - 0.09 = 2.00 - 0.09 = 1.91. Yes, that's easier. So, 3.60 minus 1.69 is 1.91. Therefore, the expression becomes:

$(x + 1.3)^2 + 1.91$

Therefore, comparing to the form $(x + b)^2 + c$, we have $b = 1.3$ and $c = 1.91$. Therefore, $b + c = 1.3 + 1.91$. Let me calculate that.

Adding 1.3 and 1.91. 1.3 is the same as 1.30. So, 1.30 + 1.91 = 3.21.

Wait, let me check again:

1.3 + 1.91.

Adding the decimal parts: 0.3 + 0.91 = 1.21, and the whole numbers: 1 + 1 = 2. Then total is 2 + 1.21 = 3.21. Yes. So, $b + c = 3.21$.

But let me verify my steps again to make sure I didn't make any mistakes.

Original quadratic: $x^2 + 2.6x + 3.6$. Completed the square by taking half of 2.6, which is 1.3, squaring it to get 1.69. Then, the expression became $(x + 1.3)^2 - 1.69 + 3.6$. Then, combining constants: -1.69 + 3.6. Which is 3.6 - 1.69. Let me confirm this calculation again.

Let me do 3.6 minus 1.69. Let me think of 3.6 as 3.60. So subtract 1.69:

Subtract 1.69 from 3.60:

Start with the tenths place: 6 tenths minus 6 tenths is 0. Then hundredths place: 0 - 9. So, need to borrow. So borrow 1 from the tenths place, making it 10 hundredths. But since the tenths place was already 6, borrowing makes it 5 tenths. Then, hundredths: 10 - 9 = 1. Then tenths: 5 tenths minus 1 tenth (because we borrowed earlier?), wait, hold on:

Wait, perhaps it's easier to subtract 1.69 from 3.60 as follows:

3.60 - 1.69 = (3.60 - 1.00) - 0.69 = 2.60 - 0.69. Then, 2.60 - 0.69. Again, subtract 0.69 from 2.60.

Which is like 2.60 - 0.60 - 0.09 = 2.00 - 0.09 = 1.91. Yes, same result. So that's correct. Therefore, c is 1.91.

Therefore, b is 1.3 and c is 1.91. Adding them together: 1.3 + 1.91.

1.3 is equal to 1.30, so adding 1.30 and 1.91:

1.30 + 1.91 = 3.21.

Therefore, the answer should be 3.21.

But let me check if there's another way to approach this, just to make sure. Maybe by expanding $(x + b)^2 + c$ and equating coefficients?

Let me try that. Let's suppose that $x^2 + 2.6x + 3.6 = (x + b)^2 + c$. Let me expand the right-hand side:

$(x + b)^2 + c = x^2 + 2bx + b^2 + c$.

So, equating coefficients with the original equation:

Original quadratic: $x^2 + 2.6x + 3.6$

Expanded form: $x^2 + 2bx + (b^2 + c)$

Therefore, equating coefficients:

For x term: 2b = 2.6 => b = 2.6 / 2 = 1.3. Which matches the previous result.

For the constant term: $b^2 + c = 3.6$. So, c = 3.6 - b^2. Since b = 1.3, then:

c = 3.6 - (1.3)^2. Which is 3.6 - 1.69 = 1.91. So same as before. Therefore, b + c = 1.3 + 1.91 = 3.21.

So that's consistent. Therefore, seems like 3.21 is the correct answer.

But just to make sure, maybe I should plug the values back into the equation and see if it's correct.

Let me compute $(x + 1.3)^2 + 1.91$. Let's expand this:

First, $(x + 1.3)^2 = x^2 + 2*1.3x + (1.3)^2 = x^2 + 2.6x + 1.69$. Then adding 1.91 gives x^2 + 2.6x + 1.69 + 1.91.

Adding 1.69 and 1.91: 1.69 + 1.91. Let me calculate that.

1.69 + 1.91 = (1 + 1) + (0.69 + 0.91) = 2 + 1.60 = 3.60.

Yes! Therefore, the expression becomes x^2 + 2.6x + 3.6, which matches the original quadratic. Perfect.

Therefore, my answer is correct.

But let me just check once more if there's any possible mistake in decimal calculations. For instance, maybe when I calculated 1.3 squared?

1.3 squared: 1.3 * 1.3. Let me do this multiplication step by step.

1.3
x1.3
--------
First, multiply 1.3 by 0.3: 0.39

Then, multiply 1.3 by 1: 1.30

Add them together: 0.39 + 1.30 = 1.69. Correct.

So that's correct.

So, no issues there.

Another thing to check is that when we completed the square, we added and subtracted 1.69. So, the original expression was x² + 2.6x + 3.6.

We added 1.69 and subtracted 1.69, so the expression becomes:

x² + 2.6x + 1.69 - 1.69 + 3.6 = (x + 1.3)^2 + ( -1.69 + 3.6 ). Which we calculated as (x + 1.3)^2 + 1.91.

Therefore, yes, that's correct.

Therefore, I can be confident that b is 1.3 and c is 1.91, so their sum is 3.21.

Alternatively, if they wanted the answer as a fraction, but the question says "as a decimal," so 3.21 is already a decimal. So, that's the answer.

Wait, just thinking, 1.3 is 13/10, 1.91 is 191/100. So adding them together:

13/10 + 191/100 = (130/100 + 191/100) = 321/100 = 3.21. Correct.

So, decimal is 3.21.

Therefore, I think that's solid. I can't find any mistakes in my reasoning.

**Final Answer**
The value of $b + c$ is \boxed{3.21}.

To rewrite the quadratic $x^2 + 2.6x + 3.6$ in the form $(x + b)^2 + c$, we will use the method of **completing the square**.

---

### Step 1: Group and identify the terms

We start with:
$$
x^2 + 2.6x + 3.6
$$

We want to express this in the form:
$$
(x + b)^2 + c
$$

This means we need to *complete the square* with respect to $x$, keeping the constant term on the side.

---

### Step 2: Complete the square

The steps to complete the square:

1. Take half of the coefficient of $x$, which is $2.6$, and square it:
   $$
   \left(\frac{2.6}{2}\right)^2 = (1.3)^2 = 1.69
   $$

2. Add and subtract this value inside the expression:
   $$
   x^2 + 2.6x + 1.69 - 1.69 + 3.6
   $$

Rewriting:
$$
(x^2 + 2.6x + 1.69) - 1.69 + 3.6
$$

Now, the trinomial $x^2 + 2.6x + 1.69$ is a perfect square:
$$
(x + 1.3)^2
$$

So the expression becomes:
$$
(x + 1.3)^2 + (3.6 - 1.69)
$$

---

### Step 3: Simplify the constant term

$$
3.6 - 1.69 = 1.91
$$

Therefore, we can write the original quadratic as:
$$
(x + 1.3)^2 + 1.91
$$

Now, comparing with $(x + b)^2 + c$, we identify:
- $b = 1.3$
- $c = 1.91$

---

### Step 4: Compute $b + c$

$$
b + c = 1.3 + 1.91 = 3.21
$$

---

### Final Answer

$$
\boxed{3.21}
$$

[Assistant]
Explanation:
Okay, so I need to rewrite the quadratic equation $x^2 + 2.6x + 3.6$ in the form $(x + b)^2 + c$, where $b$ and $c$ are constants. Then, I have to find $b + c$ as a decimal. Hmm, let me think about how to do this.

First, I remember that completing the square is the method used to convert a quadratic from standard form $ax^2 + bx + c$ to vertex form $(x + b)^2 + c$. Wait, actually, the vertex form is usually written as $a(x - h)^2 + k$, but in this case, the problem specifies $(x + b)^2 + c$, so maybe they just want it in a similar format, where the coefficient of $x^2$ is 1. Since the original quadratic has a leading coefficient of 1, that should work out.

So, let me recall the steps for completing the square. For a quadratic $x^2 + bx + c$, you take half of the coefficient of $x$, square it, and add and subtract that value to complete the square. Let me write that down.

Given $x^2 + 2.6x + 3.6$, the coefficient of $x$ is 2.6. So, half of 2.6 is... let me compute that. Half of 2.6 is 1.3. Then, squaring that gives me $1.3^2$. Let me calculate that. 1.3 times 1.3. Well, 1 times 1 is 1, 1 times 0.3 is 0.3, 0.3 times 1 is another 0.3, and 0.3 times 0.3 is 0.09. Adding those up: 1 + 0.3 + 0.3 + 0.09 = 1.69. So, $1.3^2 = 1.69$.

So, to complete the square, I need to add and subtract 1.69 inside the equation. Let me write that out:

Original expression: $x^2 + 2.6x + 3.6$

Let me rewrite it as $x^2 + 2.6x + 1.69 - 1.69 + 3.6$. Wait, but I need to make sure that adding and subtracting the same value keeps the equation equivalent. So:

$x^2 + 2.6x + 3.6 = (x^2 + 2.6x + 1.69) - 1.69 + 3.6$

Now, the first three terms form a perfect square trinomial. Let me check that. If I take $(x + 1.3)^2$, that would be $x^2 + 2*1.3x + (1.3)^2$, which is $x^2 + 2.6x + 1.69$. Yes! That's exactly what we have. So, the expression becomes:

$(x + 1.3)^2 - 1.69 + 3.6$

Now, let me compute $-1.69 + 3.6$. Let's subtract 1.69 from 3.6.

First, 3.6 minus 1.69. Let me write that as 3.60 - 1.69. Subtracting the decimals:

Starting with the hundredths place: 0 - 9. I can't do that, so I need to borrow. The tenths place is 6, so borrow 1 from the tenths place, making it 5, and the hundredths place becomes 10. Now, 10 - 9 = 1. Then, tenths place: 5 - 6. Again, can't do that, so borrow from the units place. The units place is 3, so borrow 1, making it 2, and the tenths place becomes 15. Then, 15 - 6 = 9. Finally, units place: 2 - 1 = 1. Wait, let me check again:

Wait, original numbers: 3.60 minus 1.69.

First, think of 3.60 as 3.6 and 1.69 as 1.69. So, 3.60 - 1.69. Let's subtract 1.69 from 3.60.

Alternatively, think of it as 3.60 - 1.69 = (3.60 - 1.60) - 0.09 = 2.00 - 0.09 = 1.91. Yes, that's easier. So, 3.60 minus 1.69 is 1.91. Therefore, the expression becomes:

$(x + 1.3)^2 + 1.91$

Therefore, comparing to the form $(x + b)^2 + c$, we have $b = 1.3$ and $c = 1.91$. Therefore, $b + c = 1.3 + 1.91$. Let me calculate that.

Adding 1.3 and 1.91. 1.3 is the same as 1.30. So, 1.30 + 1.91 = 3.21.

Wait, let me check again:

1.3 + 1.91.

Adding the decimal parts: 0.3 + 0.91 = 1.21, and the whole numbers: 1 + 1 = 2. Then total is 2 + 1.21 = 3.21. Yes. So, $b + c = 3.21$.

But let me verify my steps again to make sure I didn't make any mistakes.

Original quadratic: $x^2 + 2.6x + 3.6$. Completed the square by taking half of 2.6, which is 1.3, squaring it to get 1.69. Then, the expression became $(x + 1.3)^2 - 1.69 + 3.6$. Then, combining constants: -1.69 + 3.6. Which is 3.6 - 1.69. Let me confirm this calculation again.

Let me do 3.6 minus 1.69. Let me think of 3.6 as 3.60. So subtract 1.69:

Subtract 1.69 from 3.60:

Start with the tenths place: 6 tenths minus 6 tenths is 0. Then hundredths place: 0 - 9. So, need to borrow. So borrow 1 from the tenths place, making it 10 hundredths. But since the tenths place was already 6, borrowing makes it 5 tenths. Then, hundredths: 10 - 9 = 1. Then tenths: 5 tenths minus 1 tenth (because we borrowed earlier?), wait, hold on:

Wait, perhaps it's easier to subtract 1.69 from 3.60 as follows:

3.60 - 1.69 = (3.60 - 1.00) - 0.69 = 2.60 - 0.69. Then, 2.60 - 0.69. Again, subtract 0.69 from 2.60.

Which is like 2.60 - 0.60 - 0.09 = 2.00 - 0.09 = 1.91. Yes, same result. So that's correct. Therefore, c is 1.91.

Therefore, b is 1.3 and c is 1.91. Adding them together: 1.3 + 1.91.

1.3 is equal to 1.30, so adding 1.30 and 1.91:

1.30 + 1.91 = 3.21.

Therefore, the answer should be 3.21.

But let me check if there's another way to approach this, just to make sure. Maybe by expanding $(x + b)^2 + c$ and equating coefficients?

Let me try that. Let's suppose that $x^2 + 2.6x + 3.6 = (x + b)^2 + c$. Let me expand the right-hand side:

$(x + b)^2 + c = x^2 + 2bx + b^2 + c$.

So, equating coefficients with the original equation:

Original quadratic: $x^2 + 2.6x + 3.6$

Expanded form: $x^2 + 2bx + (b^2 + c)$

Therefore, equating coefficients:

For x term: 2b = 2.6 => b = 2.6 / 2 = 1.3. Which matches the previous result.

For the constant term: $b^2 + c = 3.6$. So, c = 3.6 - b^2. Since b = 1.3, then:

c = 3.6 - (1.3)^2. Which is 3.6 - 1.69 = 1.91. So same as before. Therefore, b + c = 1.3 + 1.91 = 3.21.

So that's consistent. Therefore, seems like 3.21 is the correct answer.

But just to make sure, maybe I should plug the values back into the equation and see if it's correct.

Let me compute $(x + 1.3)^2 + 1.91$. Let's expand this:

First, $(x + 1.3)^2 = x^2 + 2*1.3x + (1.3)^2 = x^2 + 2.6x + 1.69$. Then adding 1.91 gives x^2 + 2.6x + 1.69 + 1.91.

Adding 1.69 and 1.91: 1.69 + 1.91. Let me calculate that.

1.69 + 1.91 = (1 + 1) + (0.69 + 0.91) = 2 + 1.60 = 3.60.

Yes! Therefore, the expression becomes x^2 + 2.6x + 3.6, which matches the original quadratic. Perfect.

Therefore, my answer is correct.

But let me just check once more if there's any possible mistake in decimal calculations. For instance, maybe when I calculated 1.3 squared?

1.3 squared: 1.3 * 1.3. Let me do this multiplication step by step.

1.3
x1.3
--------
First, multiply 1.3 by 0.3: 0.39

Then, multiply 1.3 by 1: 1.30

Add them together: 0.39 + 1.30 = 1.69. Correct.

So that's correct.

So, no issues there.

Another thing to check is that when we completed the square, we added and subtracted 1.69. So, the original expression was x² + 2.6x + 3.6.

We added 1.69 and subtracted 1.69, so the expression becomes:

x² + 2.6x + 1.69 - 1.69 + 3.6 = (x + 1.3)^2 + ( -1.69 + 3.6 ). Which we calculated as (x + 1.3)^2 + 1.91.

Therefore, yes, that's correct.

Therefore, I can be confident that b is 1.3 and c is 1.91, so their sum is 3.21.

Alternatively, if they wanted the answer as a fraction, but the question says "as a decimal," so 3.21 is already a decimal. So, that's the answer.

Wait, just thinking, 1.3 is 13/10, 1.91 is 191/100. So adding them together:

13/10 + 191/100 = (130/100 + 191/100) = 321/100 = 3.21. Correct.

So, decimal is 3.21.

Therefore, I think that's solid. I can't find any mistakes in my reasoning.

**Final Answer**
The value of $b + c$ is \boxed{3.21}.

To rewrite the quadratic $x^2 + 2.6x + 3.6$ in the form $(x + b)^2 + c$, we will use the method of **completing the square**.

---

### Step 1: Group and identify the terms

We start with:
$$
x^2 + 2.6x + 3.6
$$

We want to express this in the form:
$$
(x + b)^2 + c
$$

This means we need to *complete the square* with respect to $x$, keeping the constant term on the side.

---

### Step 2: Complete the square

The steps to complete the square:

1. Take half of the coefficient of $x$, which is $2.6$, and square it:
   $$
   \left(\frac{2.6}{2}\right)^2 = (1.3)^2 = 1.69
   $$

2. Add and subtract this value inside the expression:
   $$
   x^2 + 2.6x + 1.69 - 1.69 + 3.6
   $$

Rewriting:
$$
(x^2 + 2.6x + 1.69) - 1.69 + 3.6
$$

Now, the trinomial $x^2 + 2.6x + 1.69$ is a perfect square:
$$
(x + 1.3)^2
$$

So the expression becomes:
$$
(x + 1.3)^2 + (3.6 - 1.69)
$$

---

### Step 3: Simplify the constant term

$$
3.6 - 1.69 = 1.91
$$

Therefore, we can write the original quadratic as:
$$
(x + 1.3)^2 + 1.91
$$

Now, comparing with $(x + b)^2 + c$, we identify:
- $b = 1.3$
- $c = 1.91$

---

### Step 4: Compute $b + c$

$$
b + c = 1.3 + 1.91 = 3.21
$$

---

### Final Answer

$$
\boxed{3.21}
$$

Solution:
We complete the square.

We have $(x+1.3)^2 = x^2 + (2.6)x + 1.69$, and so

\begin{align*}
x^2+(2.6)x+3.6 &= (x+1.3)^2 - 1.69 + 3.6 \\
&= (x+1.3)^2 + 1.91.
\end{align*}Therefore, $b=1.3$ and $c=1.91$, which gives us $b+c = \boxed{3.21}$.


Explanation:
Okay, let me try to figure out this problem. So, there's a bookcase with three shelves. The total number of books is 24. The top shelf has 8 mystery books, the middle shelf has 10 math books, and the bottom shelf has 6 science books. Then, two books are taken off each shelf. The question is asking what fraction of the remaining books are math books. Alright, let me break this down step by step.

First, let me confirm the total number of books. The problem says there are 3 shelves with a total of 24 books. Let me check if the numbers given for each shelf add up to 24. The top shelf has 8, middle has 10, and bottom has 6. Adding those up: 8 + 10 is 18, plus 6 is 24. Okay, that checks out. So, the initial counts are correct.

Now, two books are taken off each shelf. So, I need to figure out how many books are left on each shelf after removing two. Let me write that down:

Top shelf: originally 8 books, minus 2, so 8 - 2 = 6 books remaining.

Middle shelf: originally 10 math books, minus 2, so 10 - 2 = 8 books remaining.

Bottom shelf: originally 6 science books, minus 2, so 6 - 2 = 4 books remaining.

Wait, let me make sure. The problem says two books are taken off each shelf. So, each shelf loses two books, regardless of the type. So, yes, each shelf's count is reduced by two. So, the top shelf, which had mystery books, now has 6 mystery books left. The middle shelf, which had math books, now has 8 math books left. The bottom shelf, which had science books, now has 4 science books left.

But hold on, the problem says "two books are now taken off each shelf." Does this mean that two books are removed from each shelf, but we don't know which ones? Wait, but the problem is about the remaining books. However, since the question is about the fraction of math books remaining, maybe we need to consider if the books removed could be math books or not. Wait, but the problem doesn't specify which books are removed. Hmm, but the problem says "two books are now taken off each shelf." Hmm. Wait, if they just take off two books from each shelf, but we don't know if they took math books or not. But hold on, the problem mentions that the middle shelf has math books, top has mystery, and bottom has science. However, if two books are taken off each shelf, but the question is about math books. Wait, but maybe the books removed are not math books? Wait, the problem doesn't specify. So maybe we need to assume that the books removed are not math books? But that seems like an assumption. Wait, but the problem is from a math competition or something? Let me check.

Wait, actually, maybe the problem is structured so that even though two books are taken off each shelf, the math books are only on the middle shelf. So if two books are removed from each shelf, but we don't know which ones. However, the problem is asking for the fraction of math books remaining. However, since the problem mentions that the middle shelf has math books, top has mystery, bottom has science. If two books are removed from each shelf, the math books could be removed, but maybe the problem is assuming that the books removed are not math books? Wait, but that's not stated. Wait, this is confusing.

Wait, maybe the problem is designed so that the number of math books remaining is 10 - 2 = 8, assuming that two books were taken off the middle shelf, but the problem doesn't specify which books were taken. However, if you take two books off each shelf, regardless of type, the number of math books remaining would be 10 - 2 = 8, assuming that the two books taken off the middle shelf are non-math books? But that's not possible, since the middle shelf only has math books. Wait, hold on! Wait, the middle shelf has 10 math books. So, if two books are taken off the middle shelf, then those two books must be math books. Because there are no other books on the middle shelf. Similarly, the top shelf has only mystery books, so the two books taken off the top shelf are mystery books, and the bottom shelf has only science books, so the two books taken off the bottom shelf are science books.

Ah! Therefore, the total number of math books remaining is 10 - 2 = 8. Because the middle shelf only had math books, so removing two books from there would remove two math books. Similarly, the other shelves had other types of books, so removing two books from each would only affect their respective counts.

Therefore, that would mean that the number of math books remaining is 8, and the total number of books remaining is 24 - 6 = 18, since two books were removed from each shelf, so 2*3=6 books total removed.

Wait, let me verify. Total books initially: 24. Two books taken off each shelf: 3 shelves, so 2*3=6 books removed. Therefore, remaining books: 24 - 6 = 18. So total remaining books is 18.

Then, math books remaining: original math books were 10, minus 2 removed, so 8. Therefore, the fraction of math books is 8/18. Simplify that fraction. Both numbers are divisible by 2: 8 divided by 2 is 4, 18 divided by 2 is 9. So the simplified fraction is 4/9.

Wait, but let me check again. Let me make sure I didn't make a mistake here.

Original counts:

Top shelf: 8 mystery books. Remove 2, so 6 mystery books left.

Middle shelf: 10 math books. Remove 2, so 8 math books left.

Bottom shelf: 6 science books. Remove 2, so 4 science books left.

Total remaining books: 6 + 8 + 4 = 18. Correct. Total math books remaining: 8. So 8 out of 18. Which reduces to 4/9. So the answer should be 4/9?

Wait, but let me check again. Suppose that when you remove two books from each shelf, but maybe some math books are removed from other shelves? But no. The shelves are divided into types: top is mystery, middle is math, bottom is science. So there are no math books on the top or bottom shelves. Therefore, when you remove two books from the top shelf, they are mystery books, and from the bottom shelf, they are science books. Therefore, the math books only decrease by two. Therefore, math books remaining is 10 - 2 = 8. Total books remaining is 24 - 6 = 18. So 8/18 = 4/9.

Therefore, the answer is 4/9. So, is that correct? Wait, but let me make sure there isn't a trick here. The problem says "two books are now taken off each shelf." But does it specify that the books taken off are of the same type? No, it just says two books are taken off each shelf. But given that each shelf only has one type of book, then taking off two books from each shelf would mean removing two of that type. Therefore, math books would decrease by two, and other types decrease by two as well.

So, the math books are 8 out of 18 total books left. Therefore, 8/18 reduces to 4/9. Therefore, the fraction is 4/9.

Alternatively, maybe the problem is trying to trick you into thinking that math books are only on the middle shelf, but maybe some math books are on other shelves? Wait, but the problem states:

"The top shelf has 8 mystery books. The middle shelf has 10 math books. The bottom shelf has 6 science books."

So, no math books on the top or bottom shelves. So all math books are on the middle shelf, so removing two from the middle shelf would remove two math books, so remaining math books is 8. Therefore, yes, 8/18 is 4/9. Therefore, I think that is the answer.

But just to be thorough, let me consider an alternative interpretation. Suppose that the two books taken off each shelf could be any books, but the problem doesn't specify. However, since each shelf only has one type of book, the two books taken off each shelf must be of that type. So, for example, if you take two books off the top shelf, they have to be mystery books. Similarly for the others. Therefore, the counts are as we calculated. Therefore, math books remaining is 8, total books remaining is 18, so 8/18 = 4/9. Therefore, the answer is 4/9.

Alternatively, let me think if there is another way to interpret the problem. Suppose that "two books are now taken off each shelf" means that two books are removed from each shelf, but maybe they are removed in such a way that some math books are removed from other shelves? But that would be impossible because other shelves don't have math books. Therefore, that is not possible. Therefore, the only way is that math books are only on the middle shelf, so removing two from the middle shelf, so math books decrease by two. So, the answer is 4/9. So, I think that is correct. So, the answer is 4/9.

**Final Answer**
\boxed{\dfrac{4}{9}}

To solve the problem, we analyze the initial setup and then apply the changes described.

---

### Step 1: Initial Book Counts

We are told that:
- **Top shelf:** 8 mystery books
- **Middle shelf:** 10 math books
- **Bottom shelf:** 6 science books

Adding these gives a total of:
$$
8 + 10 + 6 = 24 \text{ books}
$$

---

### Step 2: Books Removed

Two books are taken off **each** shelf. So:
- **Top shelf:** 8 - 2 = 6 mystery books remaining
- **Middle shelf:** 10 - 2 = 8 math books remaining
- **Bottom shelf:** 6 - 2 = 4 science books remaining

Total books remaining:
$$
6 + 8 + 4 = 18 \text{ books}
$$

---

### Step 3: Determine the Number of Math Books Remaining

The **only** shelf with math books is the **middle shelf**, and it had 10 math books to begin with. After removing 2:
$$
10 - 2 = 8 \text{ math books remaining}
$$

---

### Step 4: Calculate the Fraction of Math Books Among Remaining Books

Now, we calculate the fraction of the remaining books that are math books:
$$
\frac{\text{Number of math books remaining}}{\text{Total number of remaining books}} = \frac{8}{18}
$$

Simplify the fraction:
$$
\frac{8}{18} = \frac{4}{9}
$$

---

### ✅ Final Answer

$$
\boxed{\dfrac{4}{9}}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, there's a bookcase with three shelves. The total number of books is 24. The top shelf has 8 mystery books, the middle shelf has 10 math books, and the bottom shelf has 6 science books. Then, two books are taken off each shelf. The question is asking what fraction of the remaining books are math books. Alright, let me break this down step by step.

First, let me confirm the total number of books. The problem says there are 3 shelves with a total of 24 books. Let me check if the numbers given for each shelf add up to 24. The top shelf has 8, middle has 10, and bottom has 6. Adding those up: 8 + 10 is 18, plus 6 is 24. Okay, that checks out. So, the initial counts are correct.

Now, two books are taken off each shelf. So, I need to figure out how many books are left on each shelf after removing two. Let me write that down:

Top shelf: originally 8 books, minus 2, so 8 - 2 = 6 books remaining.

Middle shelf: originally 10 math books, minus 2, so 10 - 2 = 8 books remaining.

Bottom shelf: originally 6 science books, minus 2, so 6 - 2 = 4 books remaining.

Wait, let me make sure. The problem says two books are taken off each shelf. So, each shelf loses two books, regardless of the type. So, yes, each shelf's count is reduced by two. So, the top shelf, which had mystery books, now has 6 mystery books left. The middle shelf, which had math books, now has 8 math books left. The bottom shelf, which had science books, now has 4 science books left.

But hold on, the problem says "two books are now taken off each shelf." Does this mean that two books are removed from each shelf, but we don't know which ones? Wait, but the problem is about the remaining books. However, since the question is about the fraction of math books remaining, maybe we need to consider if the books removed could be math books or not. Wait, but the problem doesn't specify which books are removed. Hmm, but the problem says "two books are now taken off each shelf." Hmm. Wait, if they just take off two books from each shelf, but we don't know if they took math books or not. But hold on, the problem mentions that the middle shelf has math books, top has mystery, and bottom has science. However, if two books are taken off each shelf, but the question is about math books. Wait, but maybe the books removed are not math books? Wait, the problem doesn't specify. So maybe we need to assume that the books removed are not math books? But that seems like an assumption. Wait, but the problem is from a math competition or something? Let me check.

Wait, actually, maybe the problem is structured so that even though two books are taken off each shelf, the math books are only on the middle shelf. So if two books are removed from each shelf, but we don't know which ones. However, the problem is asking for the fraction of math books remaining. However, since the problem mentions that the middle shelf has math books, top has mystery, bottom has science. If two books are removed from each shelf, the math books could be removed, but maybe the problem is assuming that the books removed are not math books? Wait, but that's not stated. Wait, this is confusing.

Wait, maybe the problem is designed so that the number of math books remaining is 10 - 2 = 8, assuming that two books were taken off the middle shelf, but the problem doesn't specify which books were taken. However, if you take two books off each shelf, regardless of type, the number of math books remaining would be 10 - 2 = 8, assuming that the two books taken off the middle shelf are non-math books? But that's not possible, since the middle shelf only has math books. Wait, hold on! Wait, the middle shelf has 10 math books. So, if two books are taken off the middle shelf, then those two books must be math books. Because there are no other books on the middle shelf. Similarly, the top shelf has only mystery books, so the two books taken off the top shelf are mystery books, and the bottom shelf has only science books, so the two books taken off the bottom shelf are science books.

Ah! Therefore, the total number of math books remaining is 10 - 2 = 8. Because the middle shelf only had math books, so removing two books from there would remove two math books. Similarly, the other shelves had other types of books, so removing two books from each would only affect their respective counts.

Therefore, that would mean that the number of math books remaining is 8, and the total number of books remaining is 24 - 6 = 18, since two books were removed from each shelf, so 2*3=6 books total removed.

Wait, let me verify. Total books initially: 24. Two books taken off each shelf: 3 shelves, so 2*3=6 books removed. Therefore, remaining books: 24 - 6 = 18. So total remaining books is 18.

Then, math books remaining: original math books were 10, minus 2 removed, so 8. Therefore, the fraction of math books is 8/18. Simplify that fraction. Both numbers are divisible by 2: 8 divided by 2 is 4, 18 divided by 2 is 9. So the simplified fraction is 4/9.

Wait, but let me check again. Let me make sure I didn't make a mistake here.

Original counts:

Top shelf: 8 mystery books. Remove 2, so 6 mystery books left.

Middle shelf: 10 math books. Remove 2, so 8 math books left.

Bottom shelf: 6 science books. Remove 2, so 4 science books left.

Total remaining books: 6 + 8 + 4 = 18. Correct. Total math books remaining: 8. So 8 out of 18. Which reduces to 4/9. So the answer should be 4/9?

Wait, but let me check again. Suppose that when you remove two books from each shelf, but maybe some math books are removed from other shelves? But no. The shelves are divided into types: top is mystery, middle is math, bottom is science. So there are no math books on the top or bottom shelves. Therefore, when you remove two books from the top shelf, they are mystery books, and from the bottom shelf, they are science books. Therefore, the math books only decrease by two. Therefore, math books remaining is 10 - 2 = 8. Total books remaining is 24 - 6 = 18. So 8/18 = 4/9.

Therefore, the answer is 4/9. So, is that correct? Wait, but let me make sure there isn't a trick here. The problem says "two books are now taken off each shelf." But does it specify that the books taken off are of the same type? No, it just says two books are taken off each shelf. But given that each shelf only has one type of book, then taking off two books from each shelf would mean removing two of that type. Therefore, math books would decrease by two, and other types decrease by two as well.

So, the math books are 8 out of 18 total books left. Therefore, 8/18 reduces to 4/9. Therefore, the fraction is 4/9.

Alternatively, maybe the problem is trying to trick you into thinking that math books are only on the middle shelf, but maybe some math books are on other shelves? Wait, but the problem states:

"The top shelf has 8 mystery books. The middle shelf has 10 math books. The bottom shelf has 6 science books."

So, no math books on the top or bottom shelves. So all math books are on the middle shelf, so removing two from the middle shelf would remove two math books, so remaining math books is 8. Therefore, yes, 8/18 is 4/9. Therefore, I think that is the answer.

But just to be thorough, let me consider an alternative interpretation. Suppose that the two books taken off each shelf could be any books, but the problem doesn't specify. However, since each shelf only has one type of book, the two books taken off each shelf must be of that type. So, for example, if you take two books off the top shelf, they have to be mystery books. Similarly for the others. Therefore, the counts are as we calculated. Therefore, math books remaining is 8, total books remaining is 18, so 8/18 = 4/9. Therefore, the answer is 4/9.

Alternatively, let me think if there is another way to interpret the problem. Suppose that "two books are now taken off each shelf" means that two books are removed from each shelf, but maybe they are removed in such a way that some math books are removed from other shelves? But that would be impossible because other shelves don't have math books. Therefore, that is not possible. Therefore, the only way is that math books are only on the middle shelf, so removing two from the middle shelf, so math books decrease by two. So, the answer is 4/9. So, I think that is correct. So, the answer is 4/9.

**Final Answer**
\boxed{\dfrac{4}{9}}

To solve the problem, we analyze the initial setup and then apply the changes described.

---

### Step 1: Initial Book Counts

We are told that:
- **Top shelf:** 8 mystery books
- **Middle shelf:** 10 math books
- **Bottom shelf:** 6 science books

Adding these gives a total of:
$$
8 + 10 + 6 = 24 \text{ books}
$$

---

### Step 2: Books Removed

Two books are taken off **each** shelf. So:
- **Top shelf:** 8 - 2 = 6 mystery books remaining
- **Middle shelf:** 10 - 2 = 8 math books remaining
- **Bottom shelf:** 6 - 2 = 4 science books remaining

Total books remaining:
$$
6 + 8 + 4 = 18 \text{ books}
$$

---

### Step 3: Determine the Number of Math Books Remaining

The **only** shelf with math books is the **middle shelf**, and it had 10 math books to begin with. After removing 2:
$$
10 - 2 = 8 \text{ math books remaining}
$$

---

### Step 4: Calculate the Fraction of Math Books Among Remaining Books

Now, we calculate the fraction of the remaining books that are math books:
$$
\frac{\text{Number of math books remaining}}{\text{Total number of remaining books}} = \frac{8}{18}
$$

Simplify the fraction:
$$
\frac{8}{18} = \frac{4}{9}
$$

---

### ✅ Final Answer

$$
\boxed{\dfrac{4}{9}}
$$

Solution: Six books are removed from the shelves, so $24-6=18$ books remain.  Of these, $10-2=8$ are math books.  Therefore, $8/18=\boxed{\frac{4}{9}}$ of the books remaining are math books.


Explanation:
Okay, so I need to find the equation of the plane that passes through the three points: (0, -1, -1), (-4, 4, 4), and (4, 5, 1). Hmm, let me recall how to do this. I think the general equation of a plane is Ax + By + Cz + D = 0, and I need to find the coefficients A, B, C, D. But there are infinitely many planes, so I need to find the specific one that goes through these three points.

First, maybe I should find two vectors that lie on the plane. Since three points define a plane, I can subtract coordinates to get vectors. Let me label the points for clarity. Let's call them P(0, -1, -1), Q(-4, 4, 4), and R(4, 5, 1).

So, vector PQ would be Q - P. Let me calculate that. The coordinates of Q minus coordinates of P: (-4 - 0, 4 - (-1), 4 - (-1)) which is (-4, 5, 5). Similarly, vector PR would be R - P. So, (4 - 0, 5 - (-1), 1 - (-1)) which is (4, 6, 2).

Now, with these two vectors, PQ and PR, I can find the normal vector to the plane by taking their cross product. The cross product of PQ and PR will give me a vector perpendicular to both, which is the normal vector of the plane.

Let me recall how to compute the cross product. If I have two vectors, say vector u = (u1, u2, u3) and vector v = (v1, v2, v3), then the cross product u × v is given by the determinant of a matrix with i, j, k in the first row, u1, u2, u3 in the second, and v1, v2, v3 in the third. So, expanding that determinant, it would be:

i*(u2v3 - u3v2) - j*(u1v3 - u3v1) + k*(u1v2 - u2v1)

So applying this to vectors PQ = (-4, 5, 5) and PR = (4, 6, 2). Let me compute each component.

First, let me write down the vectors again:

PQ = (-4, 5, 5)
PR = (4, 6, 2)

So cross product PQ × PR is:

i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 - 5*4)

Let me compute each part step by step.

For the i-component: 5*2 = 10, 5*6 = 30. So 10 - 30 = -20. So the i-component is -20.

For the j-component: It's minus [ (-4)*2 - 5*4 ] which is - [ (-8 - 20) ] = - [ -28 ] = +28. Wait, wait, hold on. Wait, the formula is - [ (u1v3 - u3v1) ]? Wait, no, maybe I need to check again.

Wait, cross product components:

The formula is:

i*(u2v3 - u3v2) - j*(u1v3 - u3v1) + k*(u1v2 - u2v1)

So for the j-component, it's negative of (u1v3 - u3v1). So in our case, u1 is -4, v3 is 2, u3 is 5, v1 is 4. So:

u1v3 - u3v1 = (-4)(2) - (5)(4) = -8 - 20 = -28. Then, the j-component is -(-28) = +28.

Then for the k-component: u1v2 - u2v1. u1 is -4, v2 is 6, u2 is 5, v1 is 4. So (-4)(6) - (5)(4) = -24 - 20 = -44.

Therefore, the cross product PQ × PR is (-20, 28, -44).

Wait, but the cross product is a vector. So this vector is perpendicular to both PQ and PR, so it's the normal vector of the plane.

Therefore, the normal vector n is (-20, 28, -44).

But I need to make sure that this is correct. Let me double-check the calculations:

First component: 5*2 - 5*6 = 10 - 30 = -20. Correct.

Second component: - [ (-4)*2 - 5*4 ] = - [ -8 - 20 ] = - [ -28 ] = +28. Correct.

Third component: (-4)*6 - 5*4 = -24 - 20 = -44. Correct.

Okay, so normal vector is (-20, 28, -44).

But maybe we can simplify this vector by dividing by common factors? Let me check if there's a common divisor. Let me compute the greatest common divisor (GCD) of 20, 28, 44.

First, factors of 20: 2, 4, 5, 10, 20.

Factors of 28: 2, 4, 7, 14, 28.

Factors of 44: 2, 4, 11, 22, 44.

So common factors are 2 and 4. So GCD is 4? Wait, 20, 28, 44. Let me check:

20 divided by 2 is 10, 28 divided by 2 is 14, 44 divided by 2 is 22. So 2 is a common divisor. Is there a higher one? Let me check if all numbers are divisible by 4: 20 ÷ 4 = 5, yes. 28 ÷ 4 = 7, yes. 44 ÷ 4 = 11, yes. So GCD is 4. Therefore, I can divide the normal vector components by 4 to simplify.

Dividing each component by 4:

-20 / 4 = -5

28 / 4 = 7

-44 / 4 = -11

So the simplified normal vector is (-5, 7, -11).

But the problem says that A, B, C, D should be integers with A > 0 and gcd(|A|, |B|, |C|, |D|) = 1. So maybe we need to take the normal vector as (-5, 7, -11), but since A has to be positive, we can multiply the entire equation by -1 to make A positive.

So if the normal vector is (-5, 7, -11), then multiplying by -1 gives (5, -7, 11). So the normal vector can be (5, -7, 11).

So then, the equation of the plane is 5x -7y + 11z + D = 0. Now, I need to find D.

To find D, I can plug in one of the points into the equation. Let me use point P(0, -1, -1).

So substituting x=0, y=-1, z=-1 into the equation:

5*(0) -7*(-1) + 11*(-1) + D = 0

Calculating:

0 + 7 -11 + D = 0

Which simplifies to:

(7 - 11) + D = 0 => -4 + D = 0 => D = 4

Wait, so D is 4? Let me check this again.

Yes:

5*0 = 0

-7*(-1) = +7

11*(-1) = -11

So total is 0 +7 -11 + D = -4 + D = 0 => D = 4.

So the equation of the plane is 5x -7y +11z +4 =0.

But let me verify this equation with another point to make sure. Let's check with point Q(-4, 4, 4).

Plugging into 5x -7y +11z +4:

5*(-4) -7*(4) +11*(4) +4 = -20 -28 +44 +4 = (-20 -28) + (44 +4) = (-48) + 48 = 0. Okay, that works.

Now check with point R(4,5,1):

5*(4) -7*(5) +11*(1) +4 = 20 -35 +11 +4 = (20 +11 +4) -35 = 35 -35 = 0. Perfect, that also works.

So seems like the equation is correct.

But let me just make sure that the coefficients have GCD 1. The coefficients are A=5, B=-7, C=11, D=4. The absolute values are 5,7,11,4. Let me check their GCD.

First, factors of 5: 1,5

Factors of 7: 1,7

Factors of 11: 1,11

Factors of 4: 1,2,4

So the only common factor is 1. Therefore, gcd is 1. So that satisfies the condition.

But the problem says A > 0. Here A is 5 which is positive, so that's good.

Therefore, the equation is 5x -7y +11z +4 = 0, which can be written as 5x -7y +11z +4 = 0.

But let me check again whether the cross product was correct, because sometimes I might have messed up signs.

Earlier, normal vector was PQ × PR = (-20, 28, -44). Then simplified by dividing by -4? Wait, no. Wait, I divided by 4, which gave (-5, 7, -11). Then multiplied by -1 to make A positive: (5, -7, 11). So yes, that is correct.

So equation is 5x -7y +11z +4 =0.

Alternatively, if I didn't simplify the normal vector, and used (-20, 28, -44), then equation would be -20x +28y -44z + D =0, then find D. Let me check with point P(0,-1,-1):

-20*0 +28*(-1) -44*(-1) + D = 0 => 0 -28 +44 + D = 16 + D = 0 => D = -16. Then equation would be -20x +28y -44z -16 = 0. Then if we divide by -4, we get 5x -7y +11z +4=0, which is the same as before. So that's consistent.

Therefore, the final equation is 5x -7y +11z +4 = 0.

But let me check once again with all points.

First, point P(0, -1, -1):

5(0) -7(-1) +11(-1) +4 = 0 +7 -11 +4 = 0. Correct.

Second, Q(-4,4,4):

5*(-4) -7*(4) +11*(4) +4 = -20 -28 +44 +4 = (-20 -28) +48 = -48 +48 = 0. Correct.

Third, R(4,5,1):

5*(4) -7*(5) +11*(1) +4 = 20 -35 +11 +4 = (20 +11 +4) -35 = 35 -35 = 0. Correct.

All points satisfy the equation, so that seems right.

But let me just check if I didn't make any mistake in cross product calculation.

Original vectors PQ = (-4,5,5) and PR = (4,6,2).

Cross product formula:

|i   j   k |
|-4 5   5|
|4   6   2|

Calculates to i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 -5*4)

Which is i*(10 -30) - j*(-8 -20) + k*(-24 -20)

Which is i*(-20) - j*(-28) + k*(-44)

Which is (-20, 28, -44). Correct.

So normal vector is correct.

Therefore, the plane equation is correct.

Hence, the answer is 5x -7y +11z +4 = 0.

But the problem says to enter it in the form Ax + By + Cz + D = 0 with A>0 and gcd(...)=1. Which we have.

So I think this is the correct answer.

**Final Answer**
The equation of the plane is \boxed{5x - 7y + 11z + 4 = 0}.

To find the equation of the plane passing through the three points
$ P(0, -1, -1) $, $ Q(-4, 4, 4) $, and $ R(4, 5, 1) $, we proceed as follows:

---

### **Step 1: Find Two Direction Vectors in the Plane**

We compute two vectors that lie on the plane by subtracting coordinates of the given points:

- **Vector $ \vec{PQ} = Q - P = (-4 - 0, 4 + 1, 4 + 1) = (-4, 5, 5) $**
- **Vector $ \vec{PR} = R - P = (4 - 0, 5 + 1, 1 + 1) = (4, 6, 2) $**

---

### **Step 2: Compute the Normal Vector via Cross Product**

The normal vector $ \vec{n} $ to the plane is found by taking the cross product of $ \vec{PQ} $ and $ \vec{PR} $.

$$
\vec{PQ} \times \vec{PR} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-4 & 5 & 5 \\
4 & 6 & 2 \\
\end{vmatrix}
= \mathbf{i}(5 \cdot 2 - 5 \cdot 6) - \mathbf{j}(-4 \cdot 2 - 5 \cdot 4) + \mathbf{k}(-4 \cdot 6 - 5 \cdot 4)
$$

Calculate each component:

- $ i $-component: $ 10 - 30 = -20 $
- $ j $-component: $ -(-8 - 20) = +28 $
- $ k $-component: $ -24 - 20 = -44 $

So the cross product gives the normal vector:
$$
\vec{n} = (-20, 28, -44)
$$

To simplify, we divide the vector by its greatest common divisor $ \gcd(20, 28, 44) = 4 $:
$$
\vec{n} = \frac{(-20, 28, -44)}{4} = (-5, 7, -11)
$$

To ensure the coefficient of $ x $ is positive (since $ A > 0 $), we multiply the entire vector by $ -1 $:
$$
\vec{n} = (5, -7, 11)
$$

---

### **Step 3: Write the General Plane Equation**

Using the normal vector $ (5, -7, 11) $, the plane equation is:

$$
5x - 7y + 11z + D = 0
$$

To find $ D $, substitute the coordinates of point $ P(0, -1, -1) $ into the equation:

$$
5(0) - 7(-1) + 11(-1) + D = 0 \\
0 + 7 - 11 + D = 0 \Rightarrow -4 + D = 0 \Rightarrow D = 4
$$

So the equation becomes:

$$
5x - 7y + 11z + 4 = 0
$$

---

### **Step 4: Verify the Equation with All Points**

- **Point $ P(0, -1, -1) $:**
  $ 5(0) - 7(-1) + 11(-1) + 4 = 0 + 7 - 11 + 4 = 0 $ ✅
- **Point $ Q(-4, 4, 4) $:**
  $ 5(-4) - 7(4) + 11(4) + 4 = -20 - 28 + 44 + 4 = 0 $ ✅
- **Point $ R(4, 5, 1) $:**
  $ 5(4) - 7(5) + 11(1) + 4 = 20 - 35 + 11 + 4 = 0 $ ✅

All points satisfy the equation.

---

### **Step 5: Ensure Conditions Are Satisfied**

- $ A = 5 > 0 $ ✅
- $ \gcd(|A|, |B|, |C|, |D|) = \gcd(5, 7, 11, 4) = 1 $ ✅
  (since 5, 7, and 11 are all primes and do not divide 4)

---

### **Final Answer**

$$
\boxed{5x - 7y + 11z + 4 = 0}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of the plane that passes through the three points: (0, -1, -1), (-4, 4, 4), and (4, 5, 1). Hmm, let me recall how to do this. I think the general equation of a plane is Ax + By + Cz + D = 0, and I need to find the coefficients A, B, C, D. But there are infinitely many planes, so I need to find the specific one that goes through these three points.

First, maybe I should find two vectors that lie on the plane. Since three points define a plane, I can subtract coordinates to get vectors. Let me label the points for clarity. Let's call them P(0, -1, -1), Q(-4, 4, 4), and R(4, 5, 1).

So, vector PQ would be Q - P. Let me calculate that. The coordinates of Q minus coordinates of P: (-4 - 0, 4 - (-1), 4 - (-1)) which is (-4, 5, 5). Similarly, vector PR would be R - P. So, (4 - 0, 5 - (-1), 1 - (-1)) which is (4, 6, 2).

Now, with these two vectors, PQ and PR, I can find the normal vector to the plane by taking their cross product. The cross product of PQ and PR will give me a vector perpendicular to both, which is the normal vector of the plane.

Let me recall how to compute the cross product. If I have two vectors, say vector u = (u1, u2, u3) and vector v = (v1, v2, v3), then the cross product u × v is given by the determinant of a matrix with i, j, k in the first row, u1, u2, u3 in the second, and v1, v2, v3 in the third. So, expanding that determinant, it would be:

i*(u2v3 - u3v2) - j*(u1v3 - u3v1) + k*(u1v2 - u2v1)

So applying this to vectors PQ = (-4, 5, 5) and PR = (4, 6, 2). Let me compute each component.

First, let me write down the vectors again:

PQ = (-4, 5, 5)
PR = (4, 6, 2)

So cross product PQ × PR is:

i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 - 5*4)

Let me compute each part step by step.

For the i-component: 5*2 = 10, 5*6 = 30. So 10 - 30 = -20. So the i-component is -20.

For the j-component: It's minus [ (-4)*2 - 5*4 ] which is - [ (-8 - 20) ] = - [ -28 ] = +28. Wait, wait, hold on. Wait, the formula is - [ (u1v3 - u3v1) ]? Wait, no, maybe I need to check again.

Wait, cross product components:

The formula is:

i*(u2v3 - u3v2) - j*(u1v3 - u3v1) + k*(u1v2 - u2v1)

So for the j-component, it's negative of (u1v3 - u3v1). So in our case, u1 is -4, v3 is 2, u3 is 5, v1 is 4. So:

u1v3 - u3v1 = (-4)(2) - (5)(4) = -8 - 20 = -28. Then, the j-component is -(-28) = +28.

Then for the k-component: u1v2 - u2v1. u1 is -4, v2 is 6, u2 is 5, v1 is 4. So (-4)(6) - (5)(4) = -24 - 20 = -44.

Therefore, the cross product PQ × PR is (-20, 28, -44).

Wait, but the cross product is a vector. So this vector is perpendicular to both PQ and PR, so it's the normal vector of the plane.

Therefore, the normal vector n is (-20, 28, -44).

But I need to make sure that this is correct. Let me double-check the calculations:

First component: 5*2 - 5*6 = 10 - 30 = -20. Correct.

Second component: - [ (-4)*2 - 5*4 ] = - [ -8 - 20 ] = - [ -28 ] = +28. Correct.

Third component: (-4)*6 - 5*4 = -24 - 20 = -44. Correct.

Okay, so normal vector is (-20, 28, -44).

But maybe we can simplify this vector by dividing by common factors? Let me check if there's a common divisor. Let me compute the greatest common divisor (GCD) of 20, 28, 44.

First, factors of 20: 2, 4, 5, 10, 20.

Factors of 28: 2, 4, 7, 14, 28.

Factors of 44: 2, 4, 11, 22, 44.

So common factors are 2 and 4. So GCD is 4? Wait, 20, 28, 44. Let me check:

20 divided by 2 is 10, 28 divided by 2 is 14, 44 divided by 2 is 22. So 2 is a common divisor. Is there a higher one? Let me check if all numbers are divisible by 4: 20 ÷ 4 = 5, yes. 28 ÷ 4 = 7, yes. 44 ÷ 4 = 11, yes. So GCD is 4. Therefore, I can divide the normal vector components by 4 to simplify.

Dividing each component by 4:

-20 / 4 = -5

28 / 4 = 7

-44 / 4 = -11

So the simplified normal vector is (-5, 7, -11).

But the problem says that A, B, C, D should be integers with A > 0 and gcd(|A|, |B|, |C|, |D|) = 1. So maybe we need to take the normal vector as (-5, 7, -11), but since A has to be positive, we can multiply the entire equation by -1 to make A positive.

So if the normal vector is (-5, 7, -11), then multiplying by -1 gives (5, -7, 11). So the normal vector can be (5, -7, 11).

So then, the equation of the plane is 5x -7y + 11z + D = 0. Now, I need to find D.

To find D, I can plug in one of the points into the equation. Let me use point P(0, -1, -1).

So substituting x=0, y=-1, z=-1 into the equation:

5*(0) -7*(-1) + 11*(-1) + D = 0

Calculating:

0 + 7 -11 + D = 0

Which simplifies to:

(7 - 11) + D = 0 => -4 + D = 0 => D = 4

Wait, so D is 4? Let me check this again.

Yes:

5*0 = 0

-7*(-1) = +7

11*(-1) = -11

So total is 0 +7 -11 + D = -4 + D = 0 => D = 4.

So the equation of the plane is 5x -7y +11z +4 =0.

But let me verify this equation with another point to make sure. Let's check with point Q(-4, 4, 4).

Plugging into 5x -7y +11z +4:

5*(-4) -7*(4) +11*(4) +4 = -20 -28 +44 +4 = (-20 -28) + (44 +4) = (-48) + 48 = 0. Okay, that works.

Now check with point R(4,5,1):

5*(4) -7*(5) +11*(1) +4 = 20 -35 +11 +4 = (20 +11 +4) -35 = 35 -35 = 0. Perfect, that also works.

So seems like the equation is correct.

But let me just make sure that the coefficients have GCD 1. The coefficients are A=5, B=-7, C=11, D=4. The absolute values are 5,7,11,4. Let me check their GCD.

First, factors of 5: 1,5

Factors of 7: 1,7

Factors of 11: 1,11

Factors of 4: 1,2,4

So the only common factor is 1. Therefore, gcd is 1. So that satisfies the condition.

But the problem says A > 0. Here A is 5 which is positive, so that's good.

Therefore, the equation is 5x -7y +11z +4 = 0, which can be written as 5x -7y +11z +4 = 0.

But let me check again whether the cross product was correct, because sometimes I might have messed up signs.

Earlier, normal vector was PQ × PR = (-20, 28, -44). Then simplified by dividing by -4? Wait, no. Wait, I divided by 4, which gave (-5, 7, -11). Then multiplied by -1 to make A positive: (5, -7, 11). So yes, that is correct.

So equation is 5x -7y +11z +4 =0.

Alternatively, if I didn't simplify the normal vector, and used (-20, 28, -44), then equation would be -20x +28y -44z + D =0, then find D. Let me check with point P(0,-1,-1):

-20*0 +28*(-1) -44*(-1) + D = 0 => 0 -28 +44 + D = 16 + D = 0 => D = -16. Then equation would be -20x +28y -44z -16 = 0. Then if we divide by -4, we get 5x -7y +11z +4=0, which is the same as before. So that's consistent.

Therefore, the final equation is 5x -7y +11z +4 = 0.

But let me check once again with all points.

First, point P(0, -1, -1):

5(0) -7(-1) +11(-1) +4 = 0 +7 -11 +4 = 0. Correct.

Second, Q(-4,4,4):

5*(-4) -7*(4) +11*(4) +4 = -20 -28 +44 +4 = (-20 -28) +48 = -48 +48 = 0. Correct.

Third, R(4,5,1):

5*(4) -7*(5) +11*(1) +4 = 20 -35 +11 +4 = (20 +11 +4) -35 = 35 -35 = 0. Correct.

All points satisfy the equation, so that seems right.

But let me just check if I didn't make any mistake in cross product calculation.

Original vectors PQ = (-4,5,5) and PR = (4,6,2).

Cross product formula:

|i   j   k |
|-4 5   5|
|4   6   2|

Calculates to i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 -5*4)

Which is i*(10 -30) - j*(-8 -20) + k*(-24 -20)

Which is i*(-20) - j*(-28) + k*(-44)

Which is (-20, 28, -44). Correct.

So normal vector is correct.

Therefore, the plane equation is correct.

Hence, the answer is 5x -7y +11z +4 = 0.

But the problem says to enter it in the form Ax + By + Cz + D = 0 with A>0 and gcd(...)=1. Which we have.

So I think this is the correct answer.

**Final Answer**
The equation of the plane is \boxed{5x - 7y + 11z + 4 = 0}.

To find the equation of the plane passing through the three points
$ P(0, -1, -1) $, $ Q(-4, 4, 4) $, and $ R(4, 5, 1) $, we proceed as follows:

---

### **Step 1: Find Two Direction Vectors in the Plane**

We compute two vectors that lie on the plane by subtracting coordinates of the given points:

- **Vector $ \vec{PQ} = Q - P = (-4 - 0, 4 + 1, 4 + 1) = (-4, 5, 5) $**
- **Vector $ \vec{PR} = R - P = (4 - 0, 5 + 1, 1 + 1) = (4, 6, 2) $**

---

### **Step 2: Compute the Normal Vector via Cross Product**

The normal vector $ \vec{n} $ to the plane is found by taking the cross product of $ \vec{PQ} $ and $ \vec{PR} $.

$$
\vec{PQ} \times \vec{PR} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-4 & 5 & 5 \\
4 & 6 & 2 \\
\end{vmatrix}
= \mathbf{i}(5 \cdot 2 - 5 \cdot 6) - \mathbf{j}(-4 \cdot 2 - 5 \cdot 4) + \mathbf{k}(-4 \cdot 6 - 5 \cdot 4)
$$

Calculate each component:

- $ i $-component: $ 10 - 30 = -20 $
- $ j $-component: $ -(-8 - 20) = +28 $
- $ k $-component: $ -24 - 20 = -44 $

So the cross product gives the normal vector:
$$
\vec{n} = (-20, 28, -44)
$$

To simplify, we divide the vector by its greatest common divisor $ \gcd(20, 28, 44) = 4 $:
$$
\vec{n} = \frac{(-20, 28, -44)}{4} = (-5, 7, -11)
$$

To ensure the coefficient of $ x $ is positive (since $ A > 0 $), we multiply the entire vector by $ -1 $:
$$
\vec{n} = (5, -7, 11)
$$

---

### **Step 3: Write the General Plane Equation**

Using the normal vector $ (5, -7, 11) $, the plane equation is:

$$
5x - 7y + 11z + D = 0
$$

To find $ D $, substitute the coordinates of point $ P(0, -1, -1) $ into the equation:

$$
5(0) - 7(-1) + 11(-1) + D = 0 \\
0 + 7 - 11 + D = 0 \Rightarrow -4 + D = 0 \Rightarrow D = 4
$$

So the equation becomes:

$$
5x - 7y + 11z + 4 = 0
$$

---

### **Step 4: Verify the Equation with All Points**

- **Point $ P(0, -1, -1) $:**
  $ 5(0) - 7(-1) + 11(-1) + 4 = 0 + 7 - 11 + 4 = 0 $ ✅
- **Point $ Q(-4, 4, 4) $:**
  $ 5(-4) - 7(4) + 11(4) + 4 = -20 - 28 + 44 + 4 = 0 $ ✅
- **Point $ R(4, 5, 1) $:**
  $ 5(4) - 7(5) + 11(1) + 4 = 20 - 35 + 11 + 4 = 0 $ ✅

All points satisfy the equation.

---

### **Step 5: Ensure Conditions Are Satisfied**

- $ A = 5 > 0 $ ✅
- $ \gcd(|A|, |B|, |C|, |D|) = \gcd(5, 7, 11, 4) = 1 $ ✅
  (since 5, 7, and 11 are all primes and do not divide 4)

---

### **Final Answer**

$$
\boxed{5x - 7y + 11z + 4 = 0}
$$

Solution:
Let $\mathbf{a} = \begin{pmatrix} 0 \\ -1 \\ -1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} -4 \\ 4 \\ 4 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 4 \\ 5 \\ 1 \end{pmatrix}.$  Then the normal vector of the plane is orthogonal to both
\[\mathbf{b} - \mathbf{a} = \begin{pmatrix} -4 \\ 5 \\ 5 \end{pmatrix}\]and
\[\mathbf{c} - \mathbf{a} = \begin{pmatrix} 4 \\ 6 \\ 2 \end{pmatrix}.\]So to compute the normal vector, we take the cross product of these vectors:
\[\begin{pmatrix} -4 \\ 5 \\ 5 \end{pmatrix} \times \begin{pmatrix} 4 \\ 6 \\ 2 \end{pmatrix} = \begin{pmatrix} -20 \\ 28 \\ -44 \end{pmatrix}.\]We can scale this vector, and take $\begin{pmatrix} 5 \\ -7 \\ 11 \end{pmatrix}$ as the normal vector.  Then the equation of the plane is of the form
\[5x - 7y + 11z + D = 0.\]Substituting the coordinates of any of the points, we find that the equation of the plane is $\boxed{5x - 7y + 11z + 4 = 0}.$