1. The problem is to solve the equation $x^2 = x + 2$ for $x$. 2. We start by rewriting the equation to set it equal to zero: $$x^2 - x - 2 = 0$$ 3. This is a quadratic equation in standard form $ax^2 + bx + c = 0$ where $a=1$, $b=-1$, and $c=-2$. 4. We can solve this using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. Substitute the values of $a$, $b$, and $c$: $$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)} = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm \sqrt{9}}{2}$$ 6. Simplify the square root: $$\sqrt{9} = 3$$ 7. So the solutions are: $$x = \frac{1 + 3}{2} = 2 \quad \text{or} \quad x = \frac{1 - 3}{2} = -1$$ 8. Therefore, the solutions to the equation $x^2 = x + 2$ are $x = 2$ and $x = -1$.