X^2 − y^2 = xy + 8

To solve the equation \( x^2 - y^2 = xy + 8 \), let's first rearrange it so that all terms are on one side:

\[
x^2 - y^2 - xy - 8 = 0
\]

Next, we can look at it as a quadratic equation in terms of \( x \):

\[
x^2 - xy - (y^2 + 8) = 0
\]

This is a quadratic equation in \( x \) that we can solve using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -y \), and \( c = -(y^2 + 8) \). Plugging in these values, we get:

\[
x = \frac{-(-y) \pm \sqrt{(-y)^2 - 4 \cdot 1 \cdot (-(y^2 + 8))}}{2 \cdot 1}
\]

This simplifies to:

\[
x = \frac{y \pm \sqrt{y^2 + 4(y^2 + 8)}}{2}
\]

\[
x = \frac{y \pm \sqrt{y^2 + 4y^2 + 32}}{2}
\]

\[
x = \frac{y \pm \sqrt{5y^2 + 32}}{2}
\]

Thus, the solutions for \( x \) in terms of \( y \) are:

\[
x = \frac{y + \sqrt{5y^2 + 32}}{2} \quad \text{and} \quad x = \frac{y - \sqrt{5y^2 + 32}}{2}
\]

These represent the two possible values of \( x \) for each given value of \( y \). If you have specific values for \( y \) in mind, we can substitute to find the corresponding values of \( x \).