Since the binary operation is defined by x y = \(x^y\), we can rewrite the given equation x 2 = 12 - x as:

\(x^2 = 12 - x\).

Rearranging this equation, we get \(x^2 + x - 12 = 0\).

This is a quadratic equation, which we can solve using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\),

a = 1, b = 1, and c = -12.

Substituting these values into the formula, we get:

\(x = \frac{-1 \pm \sqrt{1 + 48}}{2} = \frac{-1 \pm 7}{2}\).

This gives us two possible values for x: \(x = \frac{-1 + 7}{2} = 3\) and \(x = \frac{-1 - 7}{2} = -4\). Therefore, the possible values of x are 3 and -4.