Explanation:
Okay, so I need to find the distance between the foci of a hyperbola. The problem gives me the equations of the asymptotes and a point that the hyperbola passes through. Let me recall what I know about hyperbolas and their asymptotes.

First, hyperbolas have two asymptotes, which are straight lines that the hyperbola approaches but never touches. The equations of the asymptotes depend on the orientation of the hyperbola. If the hyperbola is horizontal, the asymptotes are of the form y = ±(b/a)x + k, and if it's vertical, they are y = ±(a/b)x + k, where (h, k) is the center of the hyperbola. Wait, actually, maybe I need to be more precise here.

The standard form of a hyperbola centered at (h, k) is either:

1. $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ for a horizontal hyperbola, with asymptotes y = ±(b/a)(x - h) + k.

2. $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ for a vertical hyperbola, with asymptotes y = ±(a/b)(x - h) + k.

But in this problem, the asymptotes are given as y = 2x - 3 and y = 17 - 2x. Let me note that these two lines have slopes 2 and -2. Since the slopes are positive and negative, they are symmetric with respect to the x-axis or y-axis? Wait, the slopes are 2 and -2, so they are symmetric with respect to the x-axis if the center is somewhere. But actually, the asymptotes of a hyperbola are symmetric with respect to the center. So, if the hyperbola is horizontal, the slopes of the asymptotes are ±(b/a), and if vertical, ±(a/b). So here, the slopes are 2 and -2, so they are ±2. Therefore, if the hyperbola is horizontal, then b/a = 2, so b = 2a. If it's vertical, then a/b = 2, so a = 2b. But I need to figure out which one it is. However, the equations of the asymptotes are not just y = ±(something)x, but they have intercepts. So the asymptotes are y = 2x - 3 and y = -2x + 17. So they are not passing through the origin, which means the hyperbola is not centered at the origin. Therefore, the center of the hyperbola is the intersection point of the asymptotes.

So first step: find the center of the hyperbola. The center is the intersection point of the two asymptotes. Let me solve the equations y = 2x - 3 and y = 17 - 2x.

Set 2x - 3 = 17 - 2x.

Adding 2x to both sides: 4x - 3 = 17.

Adding 3 to both sides: 4x = 20.

Divide by 4: x = 5.

Then plug back into one of the equations, say y = 2x - 3: y = 2*5 - 3 = 10 - 3 = 7. So the center is at (5, 7). Wait, that's interesting because the hyperbola passes through the point (4,7). So the center is (5,7), and the hyperbola passes through (4,7). That point is horizontally 1 unit left of the center. Let me note that.

So the center is (h, k) = (5, 7). Now, the asymptotes are y = 2x - 3 and y = -2x + 17. Let me write them in terms of the center. The asymptotes can be written as y - k = ±(slope)(x - h). Let me check.

For the first asymptote, y = 2x - 3. Let me write it as y - 7 = 2(x - 5) + (something)? Wait, let me check. If the center is (5,7), then the asymptotes should be y - 7 = ±(slope)(x - 5). Let me check if that's the case.

Take the first asymptote: y = 2x - 3. Let me rearrange it to y - 7 = 2x - 3 - 7 = 2x - 10. But 2x - 10 is 2(x - 5). So yes, y - 7 = 2(x - 5). Similarly, the other asymptote is y = -2x + 17. Let me rearrange that: y - 7 = -2x + 17 - 7 = -2x + 10 = -2(x - 5). So yes, y - 7 = -2(x - 5). Therefore, the asymptotes are y - 7 = ±2(x - 5). Therefore, the slopes are ±2, so that tells me that the hyperbola is horizontal or vertical?

Wait, for a horizontal hyperbola, the slopes of the asymptotes are ±(b/a). For vertical hyperbola, they are ±(a/b). Since the slopes here are ±2, so if it's horizontal, then b/a = 2, so b = 2a. If vertical, then a/b = 2, so a = 2b. But how do I know which one it is? Well, the standard form equations. Let me think. If the hyperbola is horizontal, then the transverse axis is horizontal, and the asymptotes are y - k = ±(b/a)(x - h). Similarly, vertical hyperbola has asymptotes y - k = ±(a/b)(x - h). So since the slopes are ±2, we need to figure out whether it's horizontal or vertical. However, without more information, maybe we can figure it out from the given point?

Wait, but maybe not. Let me think. Let me suppose that the hyperbola is horizontal. Then the equation would be [(x - 5)^2]/a^2 - [(y - 7)^2]/b^2 = 1, with b/a = 2, so b = 2a. Alternatively, if vertical, then [(y - 7)^2]/a^2 - [(x - 5)^2]/b^2 = 1, with a/b = 2, so a = 2b. So both possibilities. However, since the hyperbola passes through (4,7), which is (x, y) = (4,7). Let me plug that into both equations and see if I can find a and b.

First, let's try assuming it's a horizontal hyperbola. Then the equation is [(x - 5)^2]/a^2 - [(y - 7)^2]/b^2 = 1. Since b = 2a, substitute that in:

[(x - 5)^2]/a^2 - [(y - 7)^2]/(4a^2) = 1.

Now plug in (4,7):

[(4 - 5)^2]/a^2 - [(7 - 7)^2]/(4a^2) = 1 => [(-1)^2]/a^2 - 0 = 1 => 1/a^2 = 1 => a^2 = 1 => a = 1. Then b = 2a = 2. Therefore, the equation would be [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Let me check if this hyperbola passes through (4,7). Plugging in x=4, y=7: [(4-5)^2]/1 - [(7-7)^2]/4 = 1 - 0 = 1. Yes, that works. So that seems okay. But let me check if the vertical hyperbola also works. Let me try that.

Assume vertical hyperbola: [(y - 7)^2]/a^2 - [(x - 5)^2]/b^2 = 1. Given that the slopes of asymptotes are ±(a/b) = ±2. Therefore, a/b = 2 => a = 2b. So substitute into the equation:

[(y - 7)^2]/(4b^2) - [(x - 5)^2]/b^2 = 1.

Now plug in (4,7):

[(7 - 7)^2]/(4b^2) - [(4 - 5)^2]/b^2 = 0 - [(-1)^2]/b^2 = -1/b^2 = 1. Therefore, -1/b^2 = 1 => 1/b^2 = -1, which is impossible. Therefore, the vertical hyperbola case is invalid. Therefore, the hyperbola must be horizontal. Therefore, the equation is [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Therefore, a^2 = 1, so a = 1, and b^2 = 4, so b = 2. Then, for hyperbola, the distance between the foci is 2c, where c^2 = a^2 + b^2. Therefore, c^2 = 1 + 4 = 5, so c = sqrt(5), so distance between foci is 2*sqrt(5). Therefore, the answer is 2√5. Wait, but let me check again.

Wait, but let me make sure that I didn't make a mistake here. Let me recap. The hyperbola is horizontal, centered at (5,7), with a^2 = 1, b^2 = 4. Then c^2 = a^2 + b^2 = 1 + 4 = 5, so c = √5. Therefore, the distance between the foci is 2c = 2√5. Therefore, the answer is 2√5. But let me check if the hyperbola actually has those asymptotes. The standard asymptotes for horizontal hyperbola are y - k = ±(b/a)(x - h). Here, b/a = 2/1 = 2, so the asymptotes are y - 7 = ±2(x - 5). Which is y = 2x - 10 + 7 = 2x - 3 and y = -2x + 10 + 7 = -2x + 17. Which matches the given asymptotes. Perfect. So that's correct.

But let me check again if the point (4,7) is on the hyperbola. Plugging into [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. So [(4 - 5)^2]/1 - [(7 - 7)^2]/4 = 1 - 0 = 1. Correct. So that works. Therefore, the hyperbola is indeed [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Therefore, c^2 = 1 + 4 = 5, so c = √5, so distance between foci is 2√5. Therefore, the answer is 2√5. So boxed, that would be \boxed{2\sqrt{5}}.

Wait, but let me think again. Is there another way this could be approached? For example, sometimes hyperbola equations can be written in terms of the asymptotes. Let me recall that the equation of a hyperbola can be written as (y - k) - (2)(x - h) = 0 and (y - k) + (2)(x - h) = 0, but no, that's the asymptotes. Wait, actually, the hyperbola can be represented as (y - k - m(x - h))(y - k + m(x - h)) = c, where m is the slope? Wait, maybe not exactly. Let me think. The asymptotes are the lines that the hyperbola approaches, so if you have two lines, then the hyperbola can be represented as (y - k - m(x - h))(y - k + m(x - h)) = c, where c is a constant. But in this case, the asymptotes are y - 7 = ±2(x - 5). Therefore, the equation would be (y - 7 - 2(x - 5))(y - 7 + 2(x - 5)) = c. Let me expand that. Let me compute:

First factor: y - 7 - 2x + 10 = -2x + y + 3.

Second factor: y - 7 + 2x - 10 = 2x + y - 17.

Therefore, the equation is (-2x + y + 3)(2x + y - 17) = c. Let me multiply these two expressions. Let me note that (-2x + y + 3)(2x + y - 17) can be written as [(y + 3) - 2x][(y - 17) + 2x]. Let me expand this:

Multiply term by term:

First, (y + 3)(y - 17) = y^2 -17y + 3y -51 = y^2 -14y -51.

Then, (y + 3)(2x) = 2x(y + 3) = 2xy + 6x.

Then, (-2x)(y - 17) = -2x(y - 17) = -2xy + 34x.

Then, (-2x)(2x) = -4x^2.

Wait, actually, maybe I should do it more systematically. Let me use the formula (A - B)(A + B) where A = y + something? Wait, no. Alternatively, let me use the distributive property:

Let me denote A = -2x + y + 3 and B = 2x + y - 17. Then A*B = (-2x + y + 3)(2x + y - 17). Let me rearrange terms:

Let me write A as (y + 3) - 2x and B as (y - 17) + 2x. Then, expanding:

(y + 3)(y - 17) + (y + 3)(2x) - 2x(y - 17) - 2x*(2x). Wait, that seems complicated. Alternatively, use the formula (a + b)(c + d) = ac + ad + bc + bd. Let me do that:

First term: (-2x)(2x) = -4x²

Second term: (-2x)(y - 17) = -2x(y - 17) = -2xy + 34x

Third term: (y + 3)(2x) = 2x(y + 3) = 2xy + 6x

Fourth term: (y + 3)(y - 17) = y² -17y + 3y -51 = y² -14y -51

Now, adding all these terms together:

-4x² + (-2xy + 34x) + (2xy + 6x) + (y² -14y -51)

Combine like terms:

-4x² + (-2xy + 2xy) + (34x + 6x) + y² -14y -51

Simplify:

-4x² + 0xy + 40x + y² -14y -51

So the equation is -4x² + y² + 40x -14y -51 = c.

But this is supposed to represent the hyperbola. However, hyperbola equations are usually written in standard form. Let me rearrange this equation:

y² -4x² + 40x -14y -51 = c.

But this seems messy. However, since we know that the hyperbola passes through (4,7), we can plug that in to find c.

Let me compute the left-hand side at (4,7):

y² = 7² = 49

-4x² = -4*(16) = -64

40x = 40*4 = 160

-14y = -14*7 = -98

-51 remains.

So total: 49 -64 + 160 -98 -51.

Compute step by step:

49 -64 = -15

-15 + 160 = 145

145 -98 = 47

47 -51 = -4.

Therefore, -4 = c. Therefore, the equation of the hyperbola is -4x² + y² + 40x -14y -51 = -4. Let me rearrange this:

-4x² + y² + 40x -14y -51 + 4 = 0 => -4x² + y² + 40x -14y -47 = 0.

But this seems complicated. Let me try to write it in standard form. Let me group the x terms and y terms.

Starting with -4x² + 40x + y² -14y = 47.

Let me factor out coefficients of squared terms:

For x terms: -4x² + 40x = -4(x² - 10x)

For y terms: y² -14y = y² -14y.

Now complete the square for x and y.

Starting with x:

-4(x² - 10x). To complete the square inside the parentheses: x² -10x. Take half of -10, which is -5, square it: 25. So add and subtract 25:

-4[(x² -10x +25 -25)] = -4[(x -5)^2 -25] = -4(x -5)^2 + 100.

For y terms: y² -14y. Take half of -14, which is -7, square it: 49. Add and subtract 49:

(y² -14y +49 -49) = (y -7)^2 -49.

So putting it all back into the equation:

-4(x -5)^2 + 100 + (y -7)^2 -49 = 47.

Simplify:

-4(x -5)^2 + (y -7)^2 + 100 -49 = 47.

Which is:

-4(x -5)^2 + (y -7)^2 + 51 = 47.

Subtract 51 from both sides:

-4(x -5)^2 + (y -7)^2 = -4.

Multiply both sides by -1:

4(x -5)^2 - (y -7)^2 = 4.

Divide both sides by 4:

[(x -5)^2]/1 - [(y -7)^2]/4 = 1.

Which is exactly the equation I found earlier! So that confirms that the standard form is [(x -5)^2]/1 - [(y -7)^2]/4 = 1. Therefore, a^2 = 1, b^2 = 4, so c^2 = a^2 + b^2 = 1 + 4 = 5, so c = sqrt(5), so distance between foci is 2c = 2*sqrt(5). Therefore, the answer is 2√5. So that seems consistent.

But let me just check once more if there's another way someone might approach this problem, maybe leading to a different answer? For example, if someone thought the hyperbola was vertical? But earlier, when I tried plugging in the point (4,7) into the vertical hyperbola equation, it led to an impossible equation. So that case is invalid. Therefore, the hyperbola must be horizontal. Therefore, the answer is 2√5.

Alternatively, maybe I can think about the general equation of hyperbola with given asymptotes. The asymptotes are y = 2x - 3 and y = -2x + 17. The equation of hyperbola can be written as (y - 2x + 3)(y + 2x - 17) = k, where k is a constant. Wait, similar to what I did earlier. Let me check. If I take the two asymptotes, which are y - 2x + 3 = 0 and y + 2x - 17 = 0. Then the hyperbola equation is (y - 2x + 3)(y + 2x - 17) = k. Then, plugging in the point (4,7):

(7 - 2*4 + 3)(7 + 2*4 - 17) = (7 - 8 + 3)(7 + 8 - 17) = (2)(-2) = -4. Therefore, k = -4. Therefore, the equation is (y - 2x + 3)(y + 2x - 17) = -4. Which is the same as before. Then, expanding this equation gives the same standard form as before. Therefore, this method also leads to the same conclusion. Therefore, the answer is indeed 2√5.

Therefore, I think I can be confident that the distance between the foci is 2√5.

**Final Answer**
The distance between the foci of the hyperbola is \boxed{2\sqrt{5}}.

To find the **distance between the foci** of a hyperbola, we are given the **asymptotes** and a **point** that the hyperbola passes through. Let's proceed step by step.

---

### **Step 1: Find the Center of the Hyperbola**

The asymptotes of a hyperbola intersect at its **center**. The given asymptotes are:

$$
y = 2x - 3 \quad \text{and} \quad y = 17 - 2x
$$

To find the center, solve the system of equations:

$$
2x - 3 = 17 - 2x
$$

$$
4x = 20 \Rightarrow x = 5
$$

Substitute $ x = 5 $ into one of the asymptotes:

$$
y = 2(5) - 3 = 10 - 3 = 7
$$

So, the **center** of the hyperbola is at $ (5, 7) $.

---

### **Step 2: Determine the Orientation and Standard Form**

The slopes of the asymptotes are $ \pm 2 $, which suggests the hyperbola is **horizontal**. For a horizontal hyperbola centered at $ (h, k) $, the standard form is:

$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$

The slopes of the asymptotes are $ \pm \frac{b}{a} $, so:

$$
\frac{b}{a} = 2 \Rightarrow b = 2a
$$

---

### **Step 3: Use the Given Point to Find $ a $ and $ b $**

The hyperbola passes through the point $ (4, 7) $. Plug this into the standard form:

$$
\frac{(4 - 5)^2}{a^2} - \frac{(7 - 7)^2}{b^2} = 1
$$

$$
\frac{1}{a^2} - 0 = 1 \Rightarrow a^2 = 1 \Rightarrow a = 1
$$

Since $ b = 2a $, we get:

$$
b^2 = 4
$$

---

### **Step 4: Find the Distance Between the Foci**

For a hyperbola, the distance between the foci is $ 2c $, where:

$$
c^2 = a^2 + b^2 = 1 + 4 = 5 \Rightarrow c = \sqrt{5}
$$

So, the **distance between the foci** is:

$$
2c = 2\sqrt{5}
$$

---

### **Final Answer**

$$
\boxed{2\sqrt{5}}
$$

[Assistant]
Explanation:
Okay, so I need to find the distance between the foci of a hyperbola. The problem gives me the equations of the asymptotes and a point that the hyperbola passes through. Let me recall what I know about hyperbolas and their asymptotes.

First, hyperbolas have two asymptotes, which are straight lines that the hyperbola approaches but never touches. The equations of the asymptotes depend on the orientation of the hyperbola. If the hyperbola is horizontal, the asymptotes are of the form y = ±(b/a)x + k, and if it's vertical, they are y = ±(a/b)x + k, where (h, k) is the center of the hyperbola. Wait, actually, maybe I need to be more precise here.

The standard form of a hyperbola centered at (h, k) is either:

1. $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ for a horizontal hyperbola, with asymptotes y = ±(b/a)(x - h) + k.

2. $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ for a vertical hyperbola, with asymptotes y = ±(a/b)(x - h) + k.

But in this problem, the asymptotes are given as y = 2x - 3 and y = 17 - 2x. Let me note that these two lines have slopes 2 and -2. Since the slopes are positive and negative, they are symmetric with respect to the x-axis or y-axis? Wait, the slopes are 2 and -2, so they are symmetric with respect to the x-axis if the center is somewhere. But actually, the asymptotes of a hyperbola are symmetric with respect to the center. So, if the hyperbola is horizontal, the slopes of the asymptotes are ±(b/a), and if vertical, ±(a/b). So here, the slopes are 2 and -2, so they are ±2. Therefore, if the hyperbola is horizontal, then b/a = 2, so b = 2a. If it's vertical, then a/b = 2, so a = 2b. But I need to figure out which one it is. However, the equations of the asymptotes are not just y = ±(something)x, but they have intercepts. So the asymptotes are y = 2x - 3 and y = -2x + 17. So they are not passing through the origin, which means the hyperbola is not centered at the origin. Therefore, the center of the hyperbola is the intersection point of the asymptotes.

So first step: find the center of the hyperbola. The center is the intersection point of the two asymptotes. Let me solve the equations y = 2x - 3 and y = 17 - 2x.

Set 2x - 3 = 17 - 2x.

Adding 2x to both sides: 4x - 3 = 17.

Adding 3 to both sides: 4x = 20.

Divide by 4: x = 5.

Then plug back into one of the equations, say y = 2x - 3: y = 2*5 - 3 = 10 - 3 = 7. So the center is at (5, 7). Wait, that's interesting because the hyperbola passes through the point (4,7). So the center is (5,7), and the hyperbola passes through (4,7). That point is horizontally 1 unit left of the center. Let me note that.

So the center is (h, k) = (5, 7). Now, the asymptotes are y = 2x - 3 and y = -2x + 17. Let me write them in terms of the center. The asymptotes can be written as y - k = ±(slope)(x - h). Let me check.

For the first asymptote, y = 2x - 3. Let me write it as y - 7 = 2(x - 5) + (something)? Wait, let me check. If the center is (5,7), then the asymptotes should be y - 7 = ±(slope)(x - 5). Let me check if that's the case.

Take the first asymptote: y = 2x - 3. Let me rearrange it to y - 7 = 2x - 3 - 7 = 2x - 10. But 2x - 10 is 2(x - 5). So yes, y - 7 = 2(x - 5). Similarly, the other asymptote is y = -2x + 17. Let me rearrange that: y - 7 = -2x + 17 - 7 = -2x + 10 = -2(x - 5). So yes, y - 7 = -2(x - 5). Therefore, the asymptotes are y - 7 = ±2(x - 5). Therefore, the slopes are ±2, so that tells me that the hyperbola is horizontal or vertical?

Wait, for a horizontal hyperbola, the slopes of the asymptotes are ±(b/a). For vertical hyperbola, they are ±(a/b). Since the slopes here are ±2, so if it's horizontal, then b/a = 2, so b = 2a. If vertical, then a/b = 2, so a = 2b. But how do I know which one it is? Well, the standard form equations. Let me think. If the hyperbola is horizontal, then the transverse axis is horizontal, and the asymptotes are y - k = ±(b/a)(x - h). Similarly, vertical hyperbola has asymptotes y - k = ±(a/b)(x - h). So since the slopes are ±2, we need to figure out whether it's horizontal or vertical. However, without more information, maybe we can figure it out from the given point?

Wait, but maybe not. Let me think. Let me suppose that the hyperbola is horizontal. Then the equation would be [(x - 5)^2]/a^2 - [(y - 7)^2]/b^2 = 1, with b/a = 2, so b = 2a. Alternatively, if vertical, then [(y - 7)^2]/a^2 - [(x - 5)^2]/b^2 = 1, with a/b = 2, so a = 2b. So both possibilities. However, since the hyperbola passes through (4,7), which is (x, y) = (4,7). Let me plug that into both equations and see if I can find a and b.

First, let's try assuming it's a horizontal hyperbola. Then the equation is [(x - 5)^2]/a^2 - [(y - 7)^2]/b^2 = 1. Since b = 2a, substitute that in:

[(x - 5)^2]/a^2 - [(y - 7)^2]/(4a^2) = 1.

Now plug in (4,7):

[(4 - 5)^2]/a^2 - [(7 - 7)^2]/(4a^2) = 1 => [(-1)^2]/a^2 - 0 = 1 => 1/a^2 = 1 => a^2 = 1 => a = 1. Then b = 2a = 2. Therefore, the equation would be [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Let me check if this hyperbola passes through (4,7). Plugging in x=4, y=7: [(4-5)^2]/1 - [(7-7)^2]/4 = 1 - 0 = 1. Yes, that works. So that seems okay. But let me check if the vertical hyperbola also works. Let me try that.

Assume vertical hyperbola: [(y - 7)^2]/a^2 - [(x - 5)^2]/b^2 = 1. Given that the slopes of asymptotes are ±(a/b) = ±2. Therefore, a/b = 2 => a = 2b. So substitute into the equation:

[(y - 7)^2]/(4b^2) - [(x - 5)^2]/b^2 = 1.

Now plug in (4,7):

[(7 - 7)^2]/(4b^2) - [(4 - 5)^2]/b^2 = 0 - [(-1)^2]/b^2 = -1/b^2 = 1. Therefore, -1/b^2 = 1 => 1/b^2 = -1, which is impossible. Therefore, the vertical hyperbola case is invalid. Therefore, the hyperbola must be horizontal. Therefore, the equation is [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Therefore, a^2 = 1, so a = 1, and b^2 = 4, so b = 2. Then, for hyperbola, the distance between the foci is 2c, where c^2 = a^2 + b^2. Therefore, c^2 = 1 + 4 = 5, so c = sqrt(5), so distance between foci is 2*sqrt(5). Therefore, the answer is 2√5. Wait, but let me check again.

Wait, but let me make sure that I didn't make a mistake here. Let me recap. The hyperbola is horizontal, centered at (5,7), with a^2 = 1, b^2 = 4. Then c^2 = a^2 + b^2 = 1 + 4 = 5, so c = √5. Therefore, the distance between the foci is 2c = 2√5. Therefore, the answer is 2√5. But let me check if the hyperbola actually has those asymptotes. The standard asymptotes for horizontal hyperbola are y - k = ±(b/a)(x - h). Here, b/a = 2/1 = 2, so the asymptotes are y - 7 = ±2(x - 5). Which is y = 2x - 10 + 7 = 2x - 3 and y = -2x + 10 + 7 = -2x + 17. Which matches the given asymptotes. Perfect. So that's correct.

But let me check again if the point (4,7) is on the hyperbola. Plugging into [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. So [(4 - 5)^2]/1 - [(7 - 7)^2]/4 = 1 - 0 = 1. Correct. So that works. Therefore, the hyperbola is indeed [(x - 5)^2]/1 - [(y - 7)^2]/4 = 1. Therefore, c^2 = 1 + 4 = 5, so c = √5, so distance between foci is 2√5. Therefore, the answer is 2√5. So boxed, that would be \boxed{2\sqrt{5}}.

Wait, but let me think again. Is there another way this could be approached? For example, sometimes hyperbola equations can be written in terms of the asymptotes. Let me recall that the equation of a hyperbola can be written as (y - k) - (2)(x - h) = 0 and (y - k) + (2)(x - h) = 0, but no, that's the asymptotes. Wait, actually, the hyperbola can be represented as (y - k - m(x - h))(y - k + m(x - h)) = c, where m is the slope? Wait, maybe not exactly. Let me think. The asymptotes are the lines that the hyperbola approaches, so if you have two lines, then the hyperbola can be represented as (y - k - m(x - h))(y - k + m(x - h)) = c, where c is a constant. But in this case, the asymptotes are y - 7 = ±2(x - 5). Therefore, the equation would be (y - 7 - 2(x - 5))(y - 7 + 2(x - 5)) = c. Let me expand that. Let me compute:

First factor: y - 7 - 2x + 10 = -2x + y + 3.

Second factor: y - 7 + 2x - 10 = 2x + y - 17.

Therefore, the equation is (-2x + y + 3)(2x + y - 17) = c. Let me multiply these two expressions. Let me note that (-2x + y + 3)(2x + y - 17) can be written as [(y + 3) - 2x][(y - 17) + 2x]. Let me expand this:

Multiply term by term:

First, (y + 3)(y - 17) = y^2 -17y + 3y -51 = y^2 -14y -51.

Then, (y + 3)(2x) = 2x(y + 3) = 2xy + 6x.

Then, (-2x)(y - 17) = -2x(y - 17) = -2xy + 34x.

Then, (-2x)(2x) = -4x^2.

Wait, actually, maybe I should do it more systematically. Let me use the formula (A - B)(A + B) where A = y + something? Wait, no. Alternatively, let me use the distributive property:

Let me denote A = -2x + y + 3 and B = 2x + y - 17. Then A*B = (-2x + y + 3)(2x + y - 17). Let me rearrange terms:

Let me write A as (y + 3) - 2x and B as (y - 17) + 2x. Then, expanding:

(y + 3)(y - 17) + (y + 3)(2x) - 2x(y - 17) - 2x*(2x). Wait, that seems complicated. Alternatively, use the formula (a + b)(c + d) = ac + ad + bc + bd. Let me do that:

First term: (-2x)(2x) = -4x²

Second term: (-2x)(y - 17) = -2x(y - 17) = -2xy + 34x

Third term: (y + 3)(2x) = 2x(y + 3) = 2xy + 6x

Fourth term: (y + 3)(y - 17) = y² -17y + 3y -51 = y² -14y -51

Now, adding all these terms together:

-4x² + (-2xy + 34x) + (2xy + 6x) + (y² -14y -51)

Combine like terms:

-4x² + (-2xy + 2xy) + (34x + 6x) + y² -14y -51

Simplify:

-4x² + 0xy + 40x + y² -14y -51

So the equation is -4x² + y² + 40x -14y -51 = c.

But this is supposed to represent the hyperbola. However, hyperbola equations are usually written in standard form. Let me rearrange this equation:

y² -4x² + 40x -14y -51 = c.

But this seems messy. However, since we know that the hyperbola passes through (4,7), we can plug that in to find c.

Let me compute the left-hand side at (4,7):

y² = 7² = 49

-4x² = -4*(16) = -64

40x = 40*4 = 160

-14y = -14*7 = -98

-51 remains.

So total: 49 -64 + 160 -98 -51.

Compute step by step:

49 -64 = -15

-15 + 160 = 145

145 -98 = 47

47 -51 = -4.

Therefore, -4 = c. Therefore, the equation of the hyperbola is -4x² + y² + 40x -14y -51 = -4. Let me rearrange this:

-4x² + y² + 40x -14y -51 + 4 = 0 => -4x² + y² + 40x -14y -47 = 0.

But this seems complicated. Let me try to write it in standard form. Let me group the x terms and y terms.

Starting with -4x² + 40x + y² -14y = 47.

Let me factor out coefficients of squared terms:

For x terms: -4x² + 40x = -4(x² - 10x)

For y terms: y² -14y = y² -14y.

Now complete the square for x and y.

Starting with x:

-4(x² - 10x). To complete the square inside the parentheses: x² -10x. Take half of -10, which is -5, square it: 25. So add and subtract 25:

-4[(x² -10x +25 -25)] = -4[(x -5)^2 -25] = -4(x -5)^2 + 100.

For y terms: y² -14y. Take half of -14, which is -7, square it: 49. Add and subtract 49:

(y² -14y +49 -49) = (y -7)^2 -49.

So putting it all back into the equation:

-4(x -5)^2 + 100 + (y -7)^2 -49 = 47.

Simplify:

-4(x -5)^2 + (y -7)^2 + 100 -49 = 47.

Which is:

-4(x -5)^2 + (y -7)^2 + 51 = 47.

Subtract 51 from both sides:

-4(x -5)^2 + (y -7)^2 = -4.

Multiply both sides by -1:

4(x -5)^2 - (y -7)^2 = 4.

Divide both sides by 4:

[(x -5)^2]/1 - [(y -7)^2]/4 = 1.

Which is exactly the equation I found earlier! So that confirms that the standard form is [(x -5)^2]/1 - [(y -7)^2]/4 = 1. Therefore, a^2 = 1, b^2 = 4, so c^2 = a^2 + b^2 = 1 + 4 = 5, so c = sqrt(5), so distance between foci is 2c = 2*sqrt(5). Therefore, the answer is 2√5. So that seems consistent.

But let me just check once more if there's another way someone might approach this problem, maybe leading to a different answer? For example, if someone thought the hyperbola was vertical? But earlier, when I tried plugging in the point (4,7) into the vertical hyperbola equation, it led to an impossible equation. So that case is invalid. Therefore, the hyperbola must be horizontal. Therefore, the answer is 2√5.

Alternatively, maybe I can think about the general equation of hyperbola with given asymptotes. The asymptotes are y = 2x - 3 and y = -2x + 17. The equation of hyperbola can be written as (y - 2x + 3)(y + 2x - 17) = k, where k is a constant. Wait, similar to what I did earlier. Let me check. If I take the two asymptotes, which are y - 2x + 3 = 0 and y + 2x - 17 = 0. Then the hyperbola equation is (y - 2x + 3)(y + 2x - 17) = k. Then, plugging in the point (4,7):

(7 - 2*4 + 3)(7 + 2*4 - 17) = (7 - 8 + 3)(7 + 8 - 17) = (2)(-2) = -4. Therefore, k = -4. Therefore, the equation is (y - 2x + 3)(y + 2x - 17) = -4. Which is the same as before. Then, expanding this equation gives the same standard form as before. Therefore, this method also leads to the same conclusion. Therefore, the answer is indeed 2√5.

Therefore, I think I can be confident that the distance between the foci is 2√5.

**Final Answer**
The distance between the foci of the hyperbola is \boxed{2\sqrt{5}}.

To find the **distance between the foci** of a hyperbola, we are given the **asymptotes** and a **point** that the hyperbola passes through. Let's proceed step by step.

---

### **Step 1: Find the Center of the Hyperbola**

The asymptotes of a hyperbola intersect at its **center**. The given asymptotes are:

$$
y = 2x - 3 \quad \text{and} \quad y = 17 - 2x
$$

To find the center, solve the system of equations:

$$
2x - 3 = 17 - 2x
$$

$$
4x = 20 \Rightarrow x = 5
$$

Substitute $ x = 5 $ into one of the asymptotes:

$$
y = 2(5) - 3 = 10 - 3 = 7
$$

So, the **center** of the hyperbola is at $ (5, 7) $.

---

### **Step 2: Determine the Orientation and Standard Form**

The slopes of the asymptotes are $ \pm 2 $, which suggests the hyperbola is **horizontal**. For a horizontal hyperbola centered at $ (h, k) $, the standard form is:

$$
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
$$

The slopes of the asymptotes are $ \pm \frac{b}{a} $, so:

$$
\frac{b}{a} = 2 \Rightarrow b = 2a
$$

---

### **Step 3: Use the Given Point to Find $ a $ and $ b $**

The hyperbola passes through the point $ (4, 7) $. Plug this into the standard form:

$$
\frac{(4 - 5)^2}{a^2} - \frac{(7 - 7)^2}{b^2} = 1
$$

$$
\frac{1}{a^2} - 0 = 1 \Rightarrow a^2 = 1 \Rightarrow a = 1
$$

Since $ b = 2a $, we get:

$$
b^2 = 4
$$

---

### **Step 4: Find the Distance Between the Foci**

For a hyperbola, the distance between the foci is $ 2c $, where:

$$
c^2 = a^2 + b^2 = 1 + 4 = 5 \Rightarrow c = \sqrt{5}
$$

So, the **distance between the foci** is:

$$
2c = 2\sqrt{5}
$$

---

### **Final Answer**

$$
\boxed{2\sqrt{5}}
$$

Solution:
The intersection of the asymptotes is $(5,7),$ so this is the center of the hyperbola.  Since the slopes of the asymptotes are $\pm 2,$ the equation of the hyperbola can be written in the form
\[(x - 5)^2 - \frac{(y - 7)^2}{4} = d\]for some constant $d.$  Setting $x = 4$ and $y = 7,$ we get $d = 1,$ so the equation is
\[\frac{(x - 5)^2}{1} - \frac{(y - 7)^2}{4} = 1.\]Then $a^2 = 1$ and $b^2 = 4,$ so $c^2 = a^2 + b^2 = 5,$ which means $c = \sqrt{5}.$  Therefore, the distance between the foci is $2c = \boxed{2 \sqrt{5}}.$


Explanation:
Okay, so I need to find P(16) where P(x) is a quadratic polynomial that's squeezed between two other quadratics: x² - 2x + 2 and 2x² - 4x + 3 for all real numbers x. Also, they told me that P(11) = 181. Hmm, let me think about how to approach this.

First, since P(x) is a quadratic polynomial, I can write it in the general form: P(x) = ax² + bx + c, where a, b, c are real numbers. The problem says that for all real x, x² - 2x + 2 ≤ P(x) ≤ 2x² - 4x + 3. So, this inequality must hold for every x.

Since both the lower and upper bounds are quadratics, maybe P(x) is a quadratic that lies between them. But how can I find the coefficients a, b, c? Well, maybe I can consider the difference between P(x) and the lower bound, and the difference between the upper bound and P(x). Let me try that.

Let me denote the lower bound as L(x) = x² - 2x + 2 and the upper bound as U(x) = 2x² - 4x + 3. Then, for all x, L(x) ≤ P(x) ≤ U(x).

Since P(x) is a quadratic, maybe the difference between P(x) and L(x) is always non-negative, and the difference between U(x) and P(x) is also always non-negative. Let me write those differences:

P(x) - L(x) = (ax² + bx + c) - (x² - 2x + 2) = (a - 1)x² + (b + 2)x + (c - 2)

Similarly, U(x) - P(x) = (2x² - 4x + 3) - (ax² + bx + c) = (2 - a)x² + (-4 - b)x + (3 - c)

Since these differences must be non-negative for all real x, that means that each of these quadratics must be non-negative for all x. For a quadratic to be non-negative for all real x, it must satisfy two conditions: the leading coefficient is non-negative, and the discriminant is less than or equal to zero.

So, for P(x) - L(x) to be non-negative for all x:

1. The leading coefficient (a - 1) must be ≥ 0 ⇒ a ≥ 1
2. The discriminant must be ≤ 0. Let me compute the discriminant for P(x) - L(x):

Discriminant D1 = (b + 2)² - 4*(a - 1)*(c - 2) ≤ 0

Similarly, for U(x) - P(x) to be non-negative for all x:

1. The leading coefficient (2 - a) must be ≥ 0 ⇒ 2 - a ≥ 0 ⇒ a ≤ 2
2. The discriminant must be ≤ 0. Let me compute the discriminant for U(x) - P(x):

Discriminant D2 = (-4 - b)² - 4*(2 - a)*(3 - c) ≤ 0

So, combining the two inequalities from the leading coefficients, we have 1 ≤ a ≤ 2. Therefore, a is between 1 and 2.

Now, since P(x) is squeezed between L(x) and U(x) for all x, maybe P(x) is a linear combination of L(x) and U(x)? Or perhaps P(x) touches both bounds at some points? Wait, but since it's a quadratic, maybe it's tangent to both bounds? Or maybe it's equal to both bounds at some points?

Alternatively, since the inequalities hold for all x, maybe the difference between P(x) and L(x) is a non-negative quadratic, and the difference between U(x) and P(x) is also a non-negative quadratic. Therefore, both quadratics must be perfect squares? Or maybe they have a double root?

But maybe there's another approach. Let me think. If P(x) is between L(x) and U(x) for all x, then perhaps P(x) is equal to L(x) + something and U(x) - something else. But since both L(x) and U(x) are quadratics, maybe P(x) is a convex combination of them? Wait, but convex combination would require coefficients adding up to 1. However, since the coefficients of x² in L(x) is 1 and in U(x) is 2, so if we take a combination like P(x) = t*L(x) + (1 - t)*U(x), but then the coefficient of x² would be t*1 + (1 - t)*2 = 1 + (1 - 2t). But since a is between 1 and 2, maybe that's possible? Wait, but maybe not necessarily. Let me check.

Alternatively, since P(x) is between L(x) and U(x) for all x, then maybe P(x) is equal to L(x) + k*(U(x) - L(x)) for some constant k between 0 and 1? Let me check that. Let me compute U(x) - L(x):

U(x) - L(x) = (2x² - 4x + 3) - (x² - 2x + 2) = x² - 2x + 1 = (x - 1)^2. Oh! That's interesting. So the difference between the upper and lower bounds is a perfect square. So, U(x) - L(x) = (x - 1)^2. Therefore, if P(x) is between L(x) and U(x), then P(x) can be written as L(x) + k*(x - 1)^2, where 0 ≤ k ≤ 1. Because adding k*(x - 1)^2 to L(x) would make it lie between L(x) and U(x). Let me check:

If k = 0, then P(x) = L(x). If k = 1, then P(x) = L(x) + (x - 1)^2 = x² - 2x + 2 + x² - 2x + 1 = 2x² - 4x + 3, which is U(x). So that works. Therefore, P(x) must be of the form L(x) + k*(x - 1)^2, where k is between 0 and 1.

Alternatively, since U(x) - L(x) is (x - 1)^2, then P(x) can be written as L(x) + k*(x - 1)^2, with 0 ≤ k ≤ 1. Therefore, this would ensure that P(x) is between L(x) and U(x) for all x.

But then, since P(x) is a quadratic, let me write that out:

P(x) = L(x) + k*(x - 1)^2 = (x² - 2x + 2) + k*(x² - 2x + 1) = (1 + k)x² + (-2 - 2k)x + (2 + k)

So, P(x) = (1 + k)x² + (-2 - 2k)x + (2 + k).

Alternatively, since P(x) is a quadratic, and we have this expression, maybe we can find k using the given condition P(11) = 181. Let me try that.

So, let me compute P(11):

P(11) = (1 + k)*(11)^2 + (-2 - 2k)*(11) + (2 + k)

Let me compute each term:

First term: (1 + k)*121 = 121 + 121k

Second term: (-2 - 2k)*11 = -22 - 22k

Third term: 2 + k

Adding them all together:

121 + 121k -22 -22k + 2 + k = (121 - 22 + 2) + (121k -22k + k) = 101 + 100k

So, P(11) = 101 + 100k. But we know that P(11) = 181. Therefore:

101 + 100k = 181 ⇒ 100k = 80 ⇒ k = 80 / 100 = 0.8. So k = 4/5.

Therefore, k is 4/5, which is between 0 and 1, so that's valid.

Therefore, now we can write P(x) as:

P(x) = (1 + 4/5)x² + (-2 - 2*(4/5))x + (2 + 4/5)

Let me compute each coefficient:

First coefficient: 1 + 4/5 = 9/5

Second coefficient: -2 - 8/5 = -10/5 - 8/5 = -18/5

Third coefficient: 2 + 4/5 = 10/5 + 4/5 = 14/5

Therefore, P(x) = (9/5)x² - (18/5)x + 14/5.

Alternatively, we can factor out 1/5:

P(x) = (1/5)(9x² - 18x + 14).

Now, the question asks for P(16). Let me compute that.

First, compute 9*(16)^2 - 18*(16) + 14, then divide by 5.

Compute 16²: 256

So 9*256 = let's calculate that. 9*200 = 1800, 9*56 = 504, so total is 1800 + 504 = 2304

Then, -18*16: 18*16 = 288, so -288

Then +14.

So total is 2304 - 288 + 14. Let me compute step by step:

2304 - 288 = 2016

2016 + 14 = 2030

Therefore, P(16) = 2030 / 5 = 406.

Wait, so is that the answer? Let me check my steps again to make sure I didn't make a mistake.

First, I assumed that P(x) is of the form L(x) + k*(x - 1)^2. Since U(x) - L(x) is (x - 1)^2, which is always non-negative, so adding k*(x - 1)^2 to L(x) would give a quadratic between L(x) and U(x). That seems valid. Then, using P(11) = 181, I found k = 0.8. Then, substituting back, I found the coefficients. Then, calculating P(16) gives 406. Let me check if that's correct.

Alternatively, maybe I can check if P(x) is indeed between L(x) and U(x). Let me check for some x. For example, at x = 1, L(1) = 1 - 2 + 2 = 1, U(1) = 2 - 4 + 3 = 1. So both bounds are 1 at x = 1. Then P(1) = (9/5)(1) - (18/5)(1) + 14/5 = (9 - 18 + 14)/5 = 5/5 = 1. So P(1) = 1, which is equal to both bounds. That makes sense because the difference between U and L is (x - 1)^2, so when x = 1, the difference is zero, so P(x) must be equal to both bounds there. So that checks out.

Another check: Let me check if P(x) is indeed between L(x) and U(x). Let me take x = 0.

L(0) = 0 - 0 + 2 = 2

U(0) = 0 - 0 + 3 = 3

P(0) = 0 - 0 + 14/5 = 14/5 = 2.8, which is between 2 and 3. Good.

Another check: x = 2.

L(2) = 4 - 4 + 2 = 2

U(2) = 8 - 8 + 3 = 3

P(2) = (9/5)(4) - (18/5)(2) + 14/5 = (36/5) - (36/5) + 14/5 = 14/5 = 2.8, which is between 2 and 3. Also, since the difference between U and L is (x - 1)^2, which at x = 2 is 1, so P(x) should be L(x) + k*(1) = 2 + 0.8*1 = 2.8. Which matches. So that seems correct.

Another check: Let me check the value of P(11). According to my calculation, P(11) = 101 + 100k = 101 + 80 = 181. Which matches the given condition. So that seems correct.

Therefore, my calculation for P(16) seems correct. Let me just recheck the arithmetic for P(16):

Compute 9*(16)^2 - 18*(16) + 14.

First, 16^2 = 256. 9*256: Let me compute 256*10 = 2560, minus 256 = 2304. Correct.

Then, 18*16: 10*16=160, 8*16=128, total 288. So -288. Then +14. So 2304 - 288 = 2016, 2016 +14=2030. Divided by 5: 2030 / 5 = 406. Correct.

Therefore, the answer should be 406.

But let me think again if there's another way to approach this problem, just to confirm. For example, maybe using the fact that since P(x) is between L(x) and U(x) for all x, then the difference between P(x) and L(x) is non-negative and the difference between U(x) and P(x) is non-negative. So, if I consider the two quadratics:

P(x) - L(x) = (a - 1)x² + (b + 2)x + (c - 2) ≥ 0 for all x.

Similarly, U(x) - P(x) = (2 - a)x² + (-4 - b)x + (3 - c) ≥ 0 for all x.

Earlier, I considered that since U(x) - L(x) is a perfect square, so maybe P(x) is L(x) + k*(x - 1)^2. But maybe there's another way to find a, b, c. Let me try that approach as well, just to verify.

So, since P(x) is a quadratic, and we have inequalities for all x. Let me think about the equality cases. Since the difference between U(x) and L(x) is (x - 1)^2, which is always non-negative, and P(x) is squeezed between them. So, if P(x) is equal to L(x) + k*(x - 1)^2, then that would satisfy the inequalities for any k between 0 and 1. Therefore, that approach seems valid.

Alternatively, if I didn't think of that, maybe I could set up equations. Let me try that.

We have P(x) = ax² + bx + c.

We know that for all x, L(x) ≤ P(x) ≤ U(x).

Since both L(x) and U(x) are quadratics, and P(x) is a quadratic, the inequalities must hold for all x. Therefore, the difference P(x) - L(x) is a quadratic that is always non-negative, and U(x) - P(x) is also a quadratic that is always non-negative.

Therefore, as before, the leading coefficients must be non-negative. So, for P(x) - L(x): leading coefficient is a - 1 ≥ 0 ⇒ a ≥ 1. For U(x) - P(x): leading coefficient is 2 - a ≥ 0 ⇒ a ≤ 2. Therefore, 1 ≤ a ≤ 2.

Moreover, the discriminant of P(x) - L(x) must be ≤ 0, and the discriminant of U(x) - P(x) must be ≤ 0.

Let me write down those discriminants.

First, P(x) - L(x) = (a - 1)x² + (b + 2)x + (c - 2). Let me denote this as Q(x) = (a - 1)x² + (b + 2)x + (c - 2). For Q(x) ≥ 0 for all x, discriminant D1 ≤ 0.

Similarly, U(x) - P(x) = (2 - a)x² + (-4 - b)x + (3 - c). Let me denote this as R(x) = (2 - a)x² + (-4 - b)x + (3 - c). For R(x) ≥ 0 for all x, discriminant D2 ≤ 0.

So, D1 = (b + 2)^2 - 4*(a - 1)*(c - 2) ≤ 0

D2 = (-4 - b)^2 - 4*(2 - a)*(3 - c) ≤ 0

But this seems complicated with three variables. However, we also have the condition that P(11) = 181. So, we have another equation: a*(11)^2 + b*(11) + c = 181 ⇒ 121a + 11b + c = 181.

So, we have three equations? Wait, no. The discriminant conditions are inequalities, not equations. So, unless we can find more equations. However, since we have the equality at x = 1, because both L(x) and U(x) equal to P(x) at x = 1? Wait, earlier when I checked, P(1) = 1, which is equal to both L(1) and U(1). So, maybe P(x) touches both bounds at x = 1? Therefore, maybe x = 1 is a point where both inequalities become equalities?

If that's the case, then P(1) = L(1) = U(1) = 1. Therefore, P(1) = 1. Let me check that.

So, P(1) = a*(1)^2 + b*(1) + c = a + b + c = 1.

So, that gives another equation: a + b + c = 1.

So, now we have:

1. 121a + 11b + c = 181 (from P(11) = 181)

2. a + b + c = 1 (from P(1) = 1)

Additionally, we have the discriminant conditions. But maybe we can find more equations?

Alternatively, since we know that P(x) - L(x) is non-negative for all x, and it touches zero at x = 1 (since P(1) = L(1)), so that would mean that x = 1 is a double root of P(x) - L(x). Similarly, since P(x) - L(x) is a quadratic that is non-negative everywhere and touches zero at x = 1, then it must be a perfect square. Similarly, U(x) - P(x) is non-negative everywhere and touches zero at x = 1, so it's also a perfect square.

Wait, that seems important. If P(x) - L(x) is non-negative for all x and has a root at x = 1, then since it's a quadratic, it must have a double root at x = 1. Therefore, P(x) - L(x) = k*(x - 1)^2 for some k ≥ 0. Similarly, U(x) - P(x) = m*(x - 1)^2 for some m ≥ 0.

But since U(x) - L(x) = (x - 1)^2, then:

[P(x) - L(x)] + [U(x) - P(x)] = U(x) - L(x) = (x - 1)^2.

Therefore, k*(x - 1)^2 + m*(x - 1)^2 = (x - 1)^2 ⇒ (k + m)*(x - 1)^2 = (x - 1)^2 ⇒ k + m = 1.

Therefore, P(x) - L(x) = k*(x - 1)^2 and U(x) - P(x) = (1 - k)*(x - 1)^2.

Therefore, this gives us that P(x) = L(x) + k*(x - 1)^2, which is exactly the same as before. Therefore, this confirms the earlier approach.

Therefore, since we have P(x) = L(x) + k*(x - 1)^2, and we found k = 4/5, then P(x) is determined. Therefore, P(16) = 406.

Alternatively, if I didn't think of that, but instead used the equations:

From P(1) = 1: a + b + c = 1.

From P(11) = 181: 121a + 11b + c = 181.

Also, since P(x) - L(x) is a perfect square, which is k*(x - 1)^2. Therefore, P(x) - L(x) = k*(x - 1)^2. Therefore, expanding:

P(x) - (x² - 2x + 2) = k*(x² - 2x + 1)

Therefore, P(x) = x² - 2x + 2 + kx² - 2kx + k = (1 + k)x² + (-2 - 2k)x + (2 + k).

Which is the same as before. Therefore, we can write:

a = 1 + k

b = -2 - 2k

c = 2 + k

Therefore, from P(1) = a + b + c = (1 + k) + (-2 - 2k) + (2 + k) = 1 + k - 2 - 2k + 2 + k = (1 - 2 + 2) + (k - 2k + k) = 1 + 0 = 1. Which checks out.

Therefore, we can use the equation from P(11) = 181:

121a + 11b + c = 181.

Substituting a, b, c:

121*(1 + k) + 11*(-2 - 2k) + (2 + k) = 181

Let me compute each term:

121*(1 + k) = 121 + 121k

11*(-2 - 2k) = -22 - 22k

(2 + k) = 2 + k

Adding them together:

121 + 121k -22 -22k + 2 + k = (121 - 22 + 2) + (121k -22k + k) = 101 + 100k = 181

Which gives 100k = 80 ⇒ k = 0.8, same as before.

Therefore, this method also leads to the same result. Therefore, P(16) = 406.

Alternatively, maybe I can check if the discriminant conditions are satisfied. Let me check for P(x) - L(x) and U(x) - P(x).

First, P(x) - L(x) = k*(x - 1)^2, which is a perfect square, so discriminant is zero. Similarly, U(x) - P(x) = (1 - k)*(x - 1)^2, which is also a perfect square, so discriminant is zero. Therefore, the discriminant conditions are satisfied.

Therefore, everything checks out.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{406}

To solve the problem, we are given that $ P(x) $ is a quadratic polynomial satisfying the inequality:

$$
x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3 \quad \text{for all real } x,
$$
and that $ P(11) = 181 $. We are to find $ P(16) $.

---

### Step 1: Understand the Bounds

Let us define the lower and upper bounds as:

- $ L(x) = x^2 - 2x + 2 $
- $ U(x) = 2x^2 - 4x + 3 $

We are told that $ P(x) $ lies between these two quadratics for all real $ x $. That is:

$$
L(x) \le P(x) \le U(x) \quad \text{for all } x.
$$

Now, compute the difference between the upper and lower bounds:

$$
U(x) - L(x) = (2x^2 - 4x + 3) - (x^2 - 2x + 2) = x^2 - 2x + 1 = (x - 1)^2.
$$

This is a perfect square, which is always non-negative. Therefore, we can express $ P(x) $ as a linear combination of $ L(x) $ and $ (x - 1)^2 $, such that:

$$
P(x) = L(x) + k(x - 1)^2, \quad \text{where } 0 \le k \le 1.
$$

This ensures that $ P(x) $ lies between $ L(x) $ and $ U(x) $ for all $ x $, since $ k(x - 1)^2 $ is non-negative and $ L(x) + k(x - 1)^2 \le L(x) + (x - 1)^2 = U(x) $.

---

### Step 2: Express $ P(x) $ in Standard Form

We now write $ P(x) $ explicitly:

$$
P(x) = (x^2 - 2x + 2) + k(x^2 - 2x + 1) = (1 + k)x^2 + (-2 - 2k)x + (2 + k).
$$

So, the coefficients of $ P(x) $ are:

- $ a = 1 + k $
- $ b = -2 - 2k $
- $ c = 2 + k $

---

### Step 3: Use the Given Value $ P(11) = 181 $

Substitute $ x = 11 $ into the expression for $ P(x) $:

$$
P(11) = (1 + k)(11)^2 + (-2 - 2k)(11) + (2 + k)
$$

Compute each term:

- $ (1 + k)(121) = 121 + 121k $
- $ (-2 - 2k)(11) = -22 - 22k $
- $ 2 + k $

Add them together:

$$
P(11) = 121 + 121k - 22 - 22k + 2 + k = 101 + 100k
$$

Set this equal to 181:

$$
101 + 100k = 181 \Rightarrow 100k = 80 \Rightarrow k = \frac{4}{5}
$$

---

### Step 4: Find $ P(16) $

Now that we know $ k = \frac{4}{5} $, we can compute the coefficients of $ P(x) $:

- $ a = 1 + \frac{4}{5} = \frac{9}{5} $
- $ b = -2 - 2 \cdot \frac{4}{5} = -2 - \frac{8}{5} = -\frac{18}{5} $
- $ c = 2 + \frac{4}{5} = \frac{14}{5} $

So,

$$
P(x) = \frac{9}{5}x^2 - \frac{18}{5}x + \frac{14}{5}
$$

Now compute $ P(16) $:

$$
P(16) = \frac{9}{5}(16)^2 - \frac{18}{5}(16) + \frac{14}{5}
$$

Compute each term:

- $ 16^2 = 256 $, so $ \frac{9}{5} \cdot 256 = \frac{2304}{5} $
- $ \frac{18}{5} \cdot 16 = \frac{288}{5} $
- $ \frac{14}{5} $

Add them:

$$
P(16) = \frac{2304 - 288 + 14}{5} = \frac{2030}{5} = 406
$$

---

### Final Answer

$$
\boxed{406}
$$

[Assistant]
Explanation:
Okay, so I need to find P(16) where P(x) is a quadratic polynomial that's squeezed between two other quadratics: x² - 2x + 2 and 2x² - 4x + 3 for all real numbers x. Also, they told me that P(11) = 181. Hmm, let me think about how to approach this.

First, since P(x) is a quadratic polynomial, I can write it in the general form: P(x) = ax² + bx + c, where a, b, c are real numbers. The problem says that for all real x, x² - 2x + 2 ≤ P(x) ≤ 2x² - 4x + 3. So, this inequality must hold for every x.

Since both the lower and upper bounds are quadratics, maybe P(x) is a quadratic that lies between them. But how can I find the coefficients a, b, c? Well, maybe I can consider the difference between P(x) and the lower bound, and the difference between the upper bound and P(x). Let me try that.

Let me denote the lower bound as L(x) = x² - 2x + 2 and the upper bound as U(x) = 2x² - 4x + 3. Then, for all x, L(x) ≤ P(x) ≤ U(x).

Since P(x) is a quadratic, maybe the difference between P(x) and L(x) is always non-negative, and the difference between U(x) and P(x) is also always non-negative. Let me write those differences:

P(x) - L(x) = (ax² + bx + c) - (x² - 2x + 2) = (a - 1)x² + (b + 2)x + (c - 2)

Similarly, U(x) - P(x) = (2x² - 4x + 3) - (ax² + bx + c) = (2 - a)x² + (-4 - b)x + (3 - c)

Since these differences must be non-negative for all real x, that means that each of these quadratics must be non-negative for all x. For a quadratic to be non-negative for all real x, it must satisfy two conditions: the leading coefficient is non-negative, and the discriminant is less than or equal to zero.

So, for P(x) - L(x) to be non-negative for all x:

1. The leading coefficient (a - 1) must be ≥ 0 ⇒ a ≥ 1
2. The discriminant must be ≤ 0. Let me compute the discriminant for P(x) - L(x):

Discriminant D1 = (b + 2)² - 4*(a - 1)*(c - 2) ≤ 0

Similarly, for U(x) - P(x) to be non-negative for all x:

1. The leading coefficient (2 - a) must be ≥ 0 ⇒ 2 - a ≥ 0 ⇒ a ≤ 2
2. The discriminant must be ≤ 0. Let me compute the discriminant for U(x) - P(x):

Discriminant D2 = (-4 - b)² - 4*(2 - a)*(3 - c) ≤ 0

So, combining the two inequalities from the leading coefficients, we have 1 ≤ a ≤ 2. Therefore, a is between 1 and 2.

Now, since P(x) is squeezed between L(x) and U(x) for all x, maybe P(x) is a linear combination of L(x) and U(x)? Or perhaps P(x) touches both bounds at some points? Wait, but since it's a quadratic, maybe it's tangent to both bounds? Or maybe it's equal to both bounds at some points?

Alternatively, since the inequalities hold for all x, maybe the difference between P(x) and L(x) is a non-negative quadratic, and the difference between U(x) and P(x) is also a non-negative quadratic. Therefore, both quadratics must be perfect squares? Or maybe they have a double root?

But maybe there's another approach. Let me think. If P(x) is between L(x) and U(x) for all x, then perhaps P(x) is equal to L(x) + something and U(x) - something else. But since both L(x) and U(x) are quadratics, maybe P(x) is a convex combination of them? Wait, but convex combination would require coefficients adding up to 1. However, since the coefficients of x² in L(x) is 1 and in U(x) is 2, so if we take a combination like P(x) = t*L(x) + (1 - t)*U(x), but then the coefficient of x² would be t*1 + (1 - t)*2 = 1 + (1 - 2t). But since a is between 1 and 2, maybe that's possible? Wait, but maybe not necessarily. Let me check.

Alternatively, since P(x) is between L(x) and U(x) for all x, then maybe P(x) is equal to L(x) + k*(U(x) - L(x)) for some constant k between 0 and 1? Let me check that. Let me compute U(x) - L(x):

U(x) - L(x) = (2x² - 4x + 3) - (x² - 2x + 2) = x² - 2x + 1 = (x - 1)^2. Oh! That's interesting. So the difference between the upper and lower bounds is a perfect square. So, U(x) - L(x) = (x - 1)^2. Therefore, if P(x) is between L(x) and U(x), then P(x) can be written as L(x) + k*(x - 1)^2, where 0 ≤ k ≤ 1. Because adding k*(x - 1)^2 to L(x) would make it lie between L(x) and U(x). Let me check:

If k = 0, then P(x) = L(x). If k = 1, then P(x) = L(x) + (x - 1)^2 = x² - 2x + 2 + x² - 2x + 1 = 2x² - 4x + 3, which is U(x). So that works. Therefore, P(x) must be of the form L(x) + k*(x - 1)^2, where k is between 0 and 1.

Alternatively, since U(x) - L(x) is (x - 1)^2, then P(x) can be written as L(x) + k*(x - 1)^2, with 0 ≤ k ≤ 1. Therefore, this would ensure that P(x) is between L(x) and U(x) for all x.

But then, since P(x) is a quadratic, let me write that out:

P(x) = L(x) + k*(x - 1)^2 = (x² - 2x + 2) + k*(x² - 2x + 1) = (1 + k)x² + (-2 - 2k)x + (2 + k)

So, P(x) = (1 + k)x² + (-2 - 2k)x + (2 + k).

Alternatively, since P(x) is a quadratic, and we have this expression, maybe we can find k using the given condition P(11) = 181. Let me try that.

So, let me compute P(11):

P(11) = (1 + k)*(11)^2 + (-2 - 2k)*(11) + (2 + k)

Let me compute each term:

First term: (1 + k)*121 = 121 + 121k

Second term: (-2 - 2k)*11 = -22 - 22k

Third term: 2 + k

Adding them all together:

121 + 121k -22 -22k + 2 + k = (121 - 22 + 2) + (121k -22k + k) = 101 + 100k

So, P(11) = 101 + 100k. But we know that P(11) = 181. Therefore:

101 + 100k = 181 ⇒ 100k = 80 ⇒ k = 80 / 100 = 0.8. So k = 4/5.

Therefore, k is 4/5, which is between 0 and 1, so that's valid.

Therefore, now we can write P(x) as:

P(x) = (1 + 4/5)x² + (-2 - 2*(4/5))x + (2 + 4/5)

Let me compute each coefficient:

First coefficient: 1 + 4/5 = 9/5

Second coefficient: -2 - 8/5 = -10/5 - 8/5 = -18/5

Third coefficient: 2 + 4/5 = 10/5 + 4/5 = 14/5

Therefore, P(x) = (9/5)x² - (18/5)x + 14/5.

Alternatively, we can factor out 1/5:

P(x) = (1/5)(9x² - 18x + 14).

Now, the question asks for P(16). Let me compute that.

First, compute 9*(16)^2 - 18*(16) + 14, then divide by 5.

Compute 16²: 256

So 9*256 = let's calculate that. 9*200 = 1800, 9*56 = 504, so total is 1800 + 504 = 2304

Then, -18*16: 18*16 = 288, so -288

Then +14.

So total is 2304 - 288 + 14. Let me compute step by step:

2304 - 288 = 2016

2016 + 14 = 2030

Therefore, P(16) = 2030 / 5 = 406.

Wait, so is that the answer? Let me check my steps again to make sure I didn't make a mistake.

First, I assumed that P(x) is of the form L(x) + k*(x - 1)^2. Since U(x) - L(x) is (x - 1)^2, which is always non-negative, so adding k*(x - 1)^2 to L(x) would give a quadratic between L(x) and U(x). That seems valid. Then, using P(11) = 181, I found k = 0.8. Then, substituting back, I found the coefficients. Then, calculating P(16) gives 406. Let me check if that's correct.

Alternatively, maybe I can check if P(x) is indeed between L(x) and U(x). Let me check for some x. For example, at x = 1, L(1) = 1 - 2 + 2 = 1, U(1) = 2 - 4 + 3 = 1. So both bounds are 1 at x = 1. Then P(1) = (9/5)(1) - (18/5)(1) + 14/5 = (9 - 18 + 14)/5 = 5/5 = 1. So P(1) = 1, which is equal to both bounds. That makes sense because the difference between U and L is (x - 1)^2, so when x = 1, the difference is zero, so P(x) must be equal to both bounds there. So that checks out.

Another check: Let me check if P(x) is indeed between L(x) and U(x). Let me take x = 0.

L(0) = 0 - 0 + 2 = 2

U(0) = 0 - 0 + 3 = 3

P(0) = 0 - 0 + 14/5 = 14/5 = 2.8, which is between 2 and 3. Good.

Another check: x = 2.

L(2) = 4 - 4 + 2 = 2

U(2) = 8 - 8 + 3 = 3

P(2) = (9/5)(4) - (18/5)(2) + 14/5 = (36/5) - (36/5) + 14/5 = 14/5 = 2.8, which is between 2 and 3. Also, since the difference between U and L is (x - 1)^2, which at x = 2 is 1, so P(x) should be L(x) + k*(1) = 2 + 0.8*1 = 2.8. Which matches. So that seems correct.

Another check: Let me check the value of P(11). According to my calculation, P(11) = 101 + 100k = 101 + 80 = 181. Which matches the given condition. So that seems correct.

Therefore, my calculation for P(16) seems correct. Let me just recheck the arithmetic for P(16):

Compute 9*(16)^2 - 18*(16) + 14.

First, 16^2 = 256. 9*256: Let me compute 256*10 = 2560, minus 256 = 2304. Correct.

Then, 18*16: 10*16=160, 8*16=128, total 288. So -288. Then +14. So 2304 - 288 = 2016, 2016 +14=2030. Divided by 5: 2030 / 5 = 406. Correct.

Therefore, the answer should be 406.

But let me think again if there's another way to approach this problem, just to confirm. For example, maybe using the fact that since P(x) is between L(x) and U(x) for all x, then the difference between P(x) and L(x) is non-negative and the difference between U(x) and P(x) is non-negative. So, if I consider the two quadratics:

P(x) - L(x) = (a - 1)x² + (b + 2)x + (c - 2) ≥ 0 for all x.

Similarly, U(x) - P(x) = (2 - a)x² + (-4 - b)x + (3 - c) ≥ 0 for all x.

Earlier, I considered that since U(x) - L(x) is a perfect square, so maybe P(x) is L(x) + k*(x - 1)^2. But maybe there's another way to find a, b, c. Let me try that approach as well, just to verify.

So, since P(x) is a quadratic, and we have inequalities for all x. Let me think about the equality cases. Since the difference between U(x) and L(x) is (x - 1)^2, which is always non-negative, and P(x) is squeezed between them. So, if P(x) is equal to L(x) + k*(x - 1)^2, then that would satisfy the inequalities for any k between 0 and 1. Therefore, that approach seems valid.

Alternatively, if I didn't think of that, maybe I could set up equations. Let me try that.

We have P(x) = ax² + bx + c.

We know that for all x, L(x) ≤ P(x) ≤ U(x).

Since both L(x) and U(x) are quadratics, and P(x) is a quadratic, the inequalities must hold for all x. Therefore, the difference P(x) - L(x) is a quadratic that is always non-negative, and U(x) - P(x) is also a quadratic that is always non-negative.

Therefore, as before, the leading coefficients must be non-negative. So, for P(x) - L(x): leading coefficient is a - 1 ≥ 0 ⇒ a ≥ 1. For U(x) - P(x): leading coefficient is 2 - a ≥ 0 ⇒ a ≤ 2. Therefore, 1 ≤ a ≤ 2.

Moreover, the discriminant of P(x) - L(x) must be ≤ 0, and the discriminant of U(x) - P(x) must be ≤ 0.

Let me write down those discriminants.

First, P(x) - L(x) = (a - 1)x² + (b + 2)x + (c - 2). Let me denote this as Q(x) = (a - 1)x² + (b + 2)x + (c - 2). For Q(x) ≥ 0 for all x, discriminant D1 ≤ 0.

Similarly, U(x) - P(x) = (2 - a)x² + (-4 - b)x + (3 - c). Let me denote this as R(x) = (2 - a)x² + (-4 - b)x + (3 - c). For R(x) ≥ 0 for all x, discriminant D2 ≤ 0.

So, D1 = (b + 2)^2 - 4*(a - 1)*(c - 2) ≤ 0

D2 = (-4 - b)^2 - 4*(2 - a)*(3 - c) ≤ 0

But this seems complicated with three variables. However, we also have the condition that P(11) = 181. So, we have another equation: a*(11)^2 + b*(11) + c = 181 ⇒ 121a + 11b + c = 181.

So, we have three equations? Wait, no. The discriminant conditions are inequalities, not equations. So, unless we can find more equations. However, since we have the equality at x = 1, because both L(x) and U(x) equal to P(x) at x = 1? Wait, earlier when I checked, P(1) = 1, which is equal to both L(1) and U(1). So, maybe P(x) touches both bounds at x = 1? Therefore, maybe x = 1 is a point where both inequalities become equalities?

If that's the case, then P(1) = L(1) = U(1) = 1. Therefore, P(1) = 1. Let me check that.

So, P(1) = a*(1)^2 + b*(1) + c = a + b + c = 1.

So, that gives another equation: a + b + c = 1.

So, now we have:

1. 121a + 11b + c = 181 (from P(11) = 181)

2. a + b + c = 1 (from P(1) = 1)

Additionally, we have the discriminant conditions. But maybe we can find more equations?

Alternatively, since we know that P(x) - L(x) is non-negative for all x, and it touches zero at x = 1 (since P(1) = L(1)), so that would mean that x = 1 is a double root of P(x) - L(x). Similarly, since P(x) - L(x) is a quadratic that is non-negative everywhere and touches zero at x = 1, then it must be a perfect square. Similarly, U(x) - P(x) is non-negative everywhere and touches zero at x = 1, so it's also a perfect square.

Wait, that seems important. If P(x) - L(x) is non-negative for all x and has a root at x = 1, then since it's a quadratic, it must have a double root at x = 1. Therefore, P(x) - L(x) = k*(x - 1)^2 for some k ≥ 0. Similarly, U(x) - P(x) = m*(x - 1)^2 for some m ≥ 0.

But since U(x) - L(x) = (x - 1)^2, then:

[P(x) - L(x)] + [U(x) - P(x)] = U(x) - L(x) = (x - 1)^2.

Therefore, k*(x - 1)^2 + m*(x - 1)^2 = (x - 1)^2 ⇒ (k + m)*(x - 1)^2 = (x - 1)^2 ⇒ k + m = 1.

Therefore, P(x) - L(x) = k*(x - 1)^2 and U(x) - P(x) = (1 - k)*(x - 1)^2.

Therefore, this gives us that P(x) = L(x) + k*(x - 1)^2, which is exactly the same as before. Therefore, this confirms the earlier approach.

Therefore, since we have P(x) = L(x) + k*(x - 1)^2, and we found k = 4/5, then P(x) is determined. Therefore, P(16) = 406.

Alternatively, if I didn't think of that, but instead used the equations:

From P(1) = 1: a + b + c = 1.

From P(11) = 181: 121a + 11b + c = 181.

Also, since P(x) - L(x) is a perfect square, which is k*(x - 1)^2. Therefore, P(x) - L(x) = k*(x - 1)^2. Therefore, expanding:

P(x) - (x² - 2x + 2) = k*(x² - 2x + 1)

Therefore, P(x) = x² - 2x + 2 + kx² - 2kx + k = (1 + k)x² + (-2 - 2k)x + (2 + k).

Which is the same as before. Therefore, we can write:

a = 1 + k

b = -2 - 2k

c = 2 + k

Therefore, from P(1) = a + b + c = (1 + k) + (-2 - 2k) + (2 + k) = 1 + k - 2 - 2k + 2 + k = (1 - 2 + 2) + (k - 2k + k) = 1 + 0 = 1. Which checks out.

Therefore, we can use the equation from P(11) = 181:

121a + 11b + c = 181.

Substituting a, b, c:

121*(1 + k) + 11*(-2 - 2k) + (2 + k) = 181

Let me compute each term:

121*(1 + k) = 121 + 121k

11*(-2 - 2k) = -22 - 22k

(2 + k) = 2 + k

Adding them together:

121 + 121k -22 -22k + 2 + k = (121 - 22 + 2) + (121k -22k + k) = 101 + 100k = 181

Which gives 100k = 80 ⇒ k = 0.8, same as before.

Therefore, this method also leads to the same result. Therefore, P(16) = 406.

Alternatively, maybe I can check if the discriminant conditions are satisfied. Let me check for P(x) - L(x) and U(x) - P(x).

First, P(x) - L(x) = k*(x - 1)^2, which is a perfect square, so discriminant is zero. Similarly, U(x) - P(x) = (1 - k)*(x - 1)^2, which is also a perfect square, so discriminant is zero. Therefore, the discriminant conditions are satisfied.

Therefore, everything checks out.

Therefore, I think my answer is correct.

**Final Answer**
\boxed{406}

To solve the problem, we are given that $ P(x) $ is a quadratic polynomial satisfying the inequality:

$$
x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3 \quad \text{for all real } x,
$$
and that $ P(11) = 181 $. We are to find $ P(16) $.

---

### Step 1: Understand the Bounds

Let us define the lower and upper bounds as:

- $ L(x) = x^2 - 2x + 2 $
- $ U(x) = 2x^2 - 4x + 3 $

We are told that $ P(x) $ lies between these two quadratics for all real $ x $. That is:

$$
L(x) \le P(x) \le U(x) \quad \text{for all } x.
$$

Now, compute the difference between the upper and lower bounds:

$$
U(x) - L(x) = (2x^2 - 4x + 3) - (x^2 - 2x + 2) = x^2 - 2x + 1 = (x - 1)^2.
$$

This is a perfect square, which is always non-negative. Therefore, we can express $ P(x) $ as a linear combination of $ L(x) $ and $ (x - 1)^2 $, such that:

$$
P(x) = L(x) + k(x - 1)^2, \quad \text{where } 0 \le k \le 1.
$$

This ensures that $ P(x) $ lies between $ L(x) $ and $ U(x) $ for all $ x $, since $ k(x - 1)^2 $ is non-negative and $ L(x) + k(x - 1)^2 \le L(x) + (x - 1)^2 = U(x) $.

---

### Step 2: Express $ P(x) $ in Standard Form

We now write $ P(x) $ explicitly:

$$
P(x) = (x^2 - 2x + 2) + k(x^2 - 2x + 1) = (1 + k)x^2 + (-2 - 2k)x + (2 + k).
$$

So, the coefficients of $ P(x) $ are:

- $ a = 1 + k $
- $ b = -2 - 2k $
- $ c = 2 + k $

---

### Step 3: Use the Given Value $ P(11) = 181 $

Substitute $ x = 11 $ into the expression for $ P(x) $:

$$
P(11) = (1 + k)(11)^2 + (-2 - 2k)(11) + (2 + k)
$$

Compute each term:

- $ (1 + k)(121) = 121 + 121k $
- $ (-2 - 2k)(11) = -22 - 22k $
- $ 2 + k $

Add them together:

$$
P(11) = 121 + 121k - 22 - 22k + 2 + k = 101 + 100k
$$

Set this equal to 181:

$$
101 + 100k = 181 \Rightarrow 100k = 80 \Rightarrow k = \frac{4}{5}
$$

---

### Step 4: Find $ P(16) $

Now that we know $ k = \frac{4}{5} $, we can compute the coefficients of $ P(x) $:

- $ a = 1 + \frac{4}{5} = \frac{9}{5} $
- $ b = -2 - 2 \cdot \frac{4}{5} = -2 - \frac{8}{5} = -\frac{18}{5} $
- $ c = 2 + \frac{4}{5} = \frac{14}{5} $

So,

$$
P(x) = \frac{9}{5}x^2 - \frac{18}{5}x + \frac{14}{5}
$$

Now compute $ P(16) $:

$$
P(16) = \frac{9}{5}(16)^2 - \frac{18}{5}(16) + \frac{14}{5}
$$

Compute each term:

- $ 16^2 = 256 $, so $ \frac{9}{5} \cdot 256 = \frac{2304}{5} $
- $ \frac{18}{5} \cdot 16 = \frac{288}{5} $
- $ \frac{14}{5} $

Add them:

$$
P(16) = \frac{2304 - 288 + 14}{5} = \frac{2030}{5} = 406
$$

---

### Final Answer

$$
\boxed{406}
$$

Solution: Rewriting the given quadratics in vertex form, we have \[1 + (x-1)^2 \le P(x) \le 1 + 2(x-1)^2.\]Both of those quadratics have vertex at $(1, 1)$; considering the shape of the graph of a quadratic, we see that $P$ must also have its vertex at $(1,1)$. Therefore, \[P(x) = 1 + k(x-1)^2\]for some constant $k$. Setting $x = 11$, we have $181 = 1 +100k$, so $k = \tfrac{9}{5}$. Then \[P(16) = 1 + \tfrac{9}{5} \cdot 15^2 = \boxed{406}.\]


Explanation:
Okay, so I need to solve this equation: (3)5² - 4(5 - a)² ÷ 3 = 63. Hmm, let me make sure I understand the equation correctly. Let me rewrite it to be sure. It says 3 times 5 squared minus 4 times (5 minus a) squared divided by 3 equals 63. So, written out, it's 3*5² - [4*(5 - a)²]/3 = 63. Alright, that seems right.

First, maybe I should simplify the left side step by step. Let me start by calculating 3*5². Wait, 5 squared is 25, so 3*25 is 75. So the first term is 75. Then, the second term is 4*(5 - a)² divided by 3. So the entire equation becomes 75 - [4*(5 - a)²]/3 = 63.

Now, I need to solve for a. Let me rearrange the equation to isolate the term with a. Let me subtract 75 from both sides. That would give me - [4*(5 - a)²]/3 = 63 - 75. Calculating 63 - 75, that's -12. So now the equation is - [4*(5 - a)²]/3 = -12.

Hmm, maybe I can multiply both sides by -1 to make it simpler. That would give me [4*(5 - a)²]/3 = 12. Then, to get rid of the denominator, I can multiply both sides by 3. So 4*(5 - a)² = 36.

Now, divide both sides by 4 to solve for (5 - a)². That would be (5 - a)² = 36 / 4, which is 9. So (5 - a)² = 9.

To solve for (5 - a), take the square root of both sides. Remembering that square roots can be positive or negative, so 5 - a = ±√9, which is ±3. Therefore, 5 - a = 3 or 5 - a = -3.

Let me solve each equation separately.

First equation: 5 - a = 3. Subtract 5 from both sides: -a = 3 - 5, which is -2. Multiply both sides by -1: a = 2.

Second equation: 5 - a = -3. Subtract 5 from both sides: -a = -3 - 5, which is -8. Multiply both sides by -1: a = 8.

So the solutions are a = 2 and a = 8. The question asks for the sum of the values of a. So adding them together: 2 + 8 = 10.

Wait, let me check my steps again to make sure I didn't make any mistakes.

Original equation: 3*5² - [4*(5 - a)²]/3 = 63.

Calculating 3*25 is indeed 75. Then subtract [4*(5 - a)^2]/3 equals 63.

Subtracting 75 from both sides: - [4*(5 - a)^2]/3 = -12. Multiply both sides by -1: [4*(5 - a)^2]/3 = 12. Multiply both sides by 3: 4*(5 - a)^2 = 36. Divide by 4: (5 - a)^2 = 9. Square roots: 5 - a = ±3. Solving gives a = 5 - 3 = 2 and a = 5 - (-3) = 8.

Yes, that seems correct. Then sum is 2 + 8 = 10.

But let me plug the values back into the original equation to verify.

First, check a = 2.

Original equation: 3*5² - [4*(5 - 2)^2]/3.

Calculate 5 - 2 = 3. Then 3 squared is 9. Multiply by 4: 36. Divide by 3: 12. Then 3*25 is 75. So 75 - 12 = 63. Which matches the right side.

Now check a = 8.

Original equation: 3*5² - [4*(5 - 8)^2]/3.

5 - 8 = -3. Squared is 9. Multiply by 4: 36. Divide by 3: 12. Again, 75 - 12 = 63. Correct.

So both solutions work. Therefore, the sum is indeed 10.

I think that's all. I don't see any mistakes in my reasoning. The key steps were simplifying the equation step by step, solving the quadratic equation by taking square roots, and then verifying the solutions.

**Final Answer**
The sum of the values of $ a $ is \boxed{10}.

To solve the equation:

$$
(3)5^2 - \frac{4(5 - a)^2}{3} = 63
$$

we proceed step by step.

---

### Step 1: Simplify the known terms

First, compute $ 3 \cdot 5^2 $:

$$
3 \cdot 25 = 75
$$

So the equation becomes:

$$
75 - \frac{4(5 - a)^2}{3} = 63
$$

---

### Step 2: Isolate the term with $ a $

Subtract 75 from both sides:

$$
- \frac{4(5 - a)^2}{3} = 63 - 75 = -12
$$

Multiply both sides by -1:

$$
\frac{4(5 - a)^2}{3} = 12
$$

---

### Step 3: Eliminate the denominator

Multiply both sides by 3:

$$
4(5 - a)^2 = 36
$$

Divide both sides by 4:

$$
(5 - a)^2 = 9
$$

---

### Step 4: Solve the quadratic equation

Take the square root of both sides:

$$
5 - a = \pm 3
$$

This gives two equations:

1. $ 5 - a = 3 \Rightarrow a = 2 $
2. $ 5 - a = -3 \Rightarrow a = 8 $

---

### Step 5: Find the sum of the solutions

The two values of $ a $ are 2 and 8. Their sum is:

$$
2 + 8 = 10
$$

---

### Final Answer

$$
\boxed{10}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation: (3)5² - 4(5 - a)² ÷ 3 = 63. Hmm, let me make sure I understand the equation correctly. Let me rewrite it to be sure. It says 3 times 5 squared minus 4 times (5 minus a) squared divided by 3 equals 63. So, written out, it's 3*5² - [4*(5 - a)²]/3 = 63. Alright, that seems right.

First, maybe I should simplify the left side step by step. Let me start by calculating 3*5². Wait, 5 squared is 25, so 3*25 is 75. So the first term is 75. Then, the second term is 4*(5 - a)² divided by 3. So the entire equation becomes 75 - [4*(5 - a)²]/3 = 63.

Now, I need to solve for a. Let me rearrange the equation to isolate the term with a. Let me subtract 75 from both sides. That would give me - [4*(5 - a)²]/3 = 63 - 75. Calculating 63 - 75, that's -12. So now the equation is - [4*(5 - a)²]/3 = -12.

Hmm, maybe I can multiply both sides by -1 to make it simpler. That would give me [4*(5 - a)²]/3 = 12. Then, to get rid of the denominator, I can multiply both sides by 3. So 4*(5 - a)² = 36.

Now, divide both sides by 4 to solve for (5 - a)². That would be (5 - a)² = 36 / 4, which is 9. So (5 - a)² = 9.

To solve for (5 - a), take the square root of both sides. Remembering that square roots can be positive or negative, so 5 - a = ±√9, which is ±3. Therefore, 5 - a = 3 or 5 - a = -3.

Let me solve each equation separately.

First equation: 5 - a = 3. Subtract 5 from both sides: -a = 3 - 5, which is -2. Multiply both sides by -1: a = 2.

Second equation: 5 - a = -3. Subtract 5 from both sides: -a = -3 - 5, which is -8. Multiply both sides by -1: a = 8.

So the solutions are a = 2 and a = 8. The question asks for the sum of the values of a. So adding them together: 2 + 8 = 10.

Wait, let me check my steps again to make sure I didn't make any mistakes.

Original equation: 3*5² - [4*(5 - a)²]/3 = 63.

Calculating 3*25 is indeed 75. Then subtract [4*(5 - a)^2]/3 equals 63.

Subtracting 75 from both sides: - [4*(5 - a)^2]/3 = -12. Multiply both sides by -1: [4*(5 - a)^2]/3 = 12. Multiply both sides by 3: 4*(5 - a)^2 = 36. Divide by 4: (5 - a)^2 = 9. Square roots: 5 - a = ±3. Solving gives a = 5 - 3 = 2 and a = 5 - (-3) = 8.

Yes, that seems correct. Then sum is 2 + 8 = 10.

But let me plug the values back into the original equation to verify.

First, check a = 2.

Original equation: 3*5² - [4*(5 - 2)^2]/3.

Calculate 5 - 2 = 3. Then 3 squared is 9. Multiply by 4: 36. Divide by 3: 12. Then 3*25 is 75. So 75 - 12 = 63. Which matches the right side.

Now check a = 8.

Original equation: 3*5² - [4*(5 - 8)^2]/3.

5 - 8 = -3. Squared is 9. Multiply by 4: 36. Divide by 3: 12. Again, 75 - 12 = 63. Correct.

So both solutions work. Therefore, the sum is indeed 10.

I think that's all. I don't see any mistakes in my reasoning. The key steps were simplifying the equation step by step, solving the quadratic equation by taking square roots, and then verifying the solutions.

**Final Answer**
The sum of the values of $ a $ is \boxed{10}.

To solve the equation:

$$
(3)5^2 - \frac{4(5 - a)^2}{3} = 63
$$

we proceed step by step.

---

### Step 1: Simplify the known terms

First, compute $ 3 \cdot 5^2 $:

$$
3 \cdot 25 = 75
$$

So the equation becomes:

$$
75 - \frac{4(5 - a)^2}{3} = 63
$$

---

### Step 2: Isolate the term with $ a $

Subtract 75 from both sides:

$$
- \frac{4(5 - a)^2}{3} = 63 - 75 = -12
$$

Multiply both sides by -1:

$$
\frac{4(5 - a)^2}{3} = 12
$$

---

### Step 3: Eliminate the denominator

Multiply both sides by 3:

$$
4(5 - a)^2 = 36
$$

Divide both sides by 4:

$$
(5 - a)^2 = 9
$$

---

### Step 4: Solve the quadratic equation

Take the square root of both sides:

$$
5 - a = \pm 3
$$

This gives two equations:

1. $ 5 - a = 3 \Rightarrow a = 2 $
2. $ 5 - a = -3 \Rightarrow a = 8 $

---

### Step 5: Find the sum of the solutions

The two values of $ a $ are 2 and 8. Their sum is:

$$
2 + 8 = 10
$$

---

### Final Answer

$$
\boxed{10}
$$

Solution:
First we multiply all terms in the equation by 3 to avoid fractions and then solve for $a$. \begin{align*}
9\cdot5^2-4(5-a)^2&=3\cdot63\quad\Rightarrow\\
-4(5-a)^2&=9\cdot21-9\cdot25\quad\Rightarrow\\
&=9(-4)\quad\Rightarrow\\
(5-a)^2&=9
\end{align*} Therefore, \begin{align*}
5-a=3\quad\text{ OR }\quad 5-a=-3\quad\Rightarrow\\
2=a \quad\text{ OR }\quad 8=a.
\end{align*} The sum of the values of $a$ is $2+8=\boxed{10}$.