If $(x,y)$ is a solution to the system
\begin{align*}
xy &= 6, \\
x^2 y + xy^2 + x + y &= 63,
\end{align*}find $x^2 + y^2.$
The second equation factors as $(xy + 1)(x + y) = 63,$ so $7(x + y) = 63,$ or $x + y = 9.$  Then
\[x^2 + y^2 = (x + y)^2 - 2xy = 9^2 - 2 \cdot 6 = \boxed{69}.\]
69
Level 2
Intermediate Algebra
Problem: If $(x,y)$ is a solution to the system
\begin{align*}
xy &= 6, \\
x^2 y + xy^2 + x + y &= 63,
\end{align*}find $x^2 + y^2.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find $x^2 + y^2$, we start with the equations given. The first equation, $xy = 6$, allows us to express $y$ in terms of $x$: $y = \frac{6}{x}$. 

Next, we substitute $y$ into the second equation, $x^2 y + xy^2 + x + y = 63$. Replacing $y$ gives us:

$x^2 \left(\frac{6}{x}\right) + x \left(\frac{6}{x}\right)^2 + x + \frac{6}{x} = 63$.

This simplifies to:

$6x + \frac{36}{x} + x + \frac{6}{x} = 63$.

Combining like terms results in:

$7x + \frac{42}{x} = 63$.

To eliminate the fraction, we multiply through by $x$ (assuming $x \neq 0$):

$7x^2 + 42 = 63x$.

Rearranging gives us a standard quadratic equation:

$7x^2 - 63x + 42 = 0$.

Dividing the entire equation by 7 simplifies it to:

$x^2 - 9x + 6 = 0$.

Next, we apply the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -9$, and $c = 6$:

$x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} = \frac{9 \pm \sqrt{81 - 24}}{2} = \frac{9 \pm \sqrt{57}}{2}$.

This gives us two possible values for $x$. We can find the corresponding values for $y$ using $y = \frac{6}{x}$.

Next, we need to calculate $x^2 + y^2$. We use the identity $x^2 + y^2 = (x+y)^2 - 2xy$. 

First, we find $x + y$:

$x + y = x + \frac{6}{x} = \frac{x^2 + 6}{x}$.

Now we calculate $xy = 6$. 

Next, we compute $(x+y)^2$:

$(x+y)^2 = \left(\frac{x^2 + 6}{x}\right)^2 = \frac{(x^2 + 6)^2}{x^2}$.

Now we substitute into the identity:

$x^2 + y^2 = \frac{(x^2 + 6)^2}{x^2} - 2 \cdot 6$.

After simplifying, we find that $x^2 + y^2$ evaluates to $69$. 

Thus, the final answer is $69$.<|END_OF_LATENT|> The final answer is $69$.