How to get the critical values

$$f(x) = g(x)$$ $$x^2 = x+2$$

Subtract $x$ and 2 from both sides:

$$x^2 - x - 2 = 0$$

You can solve this by factoring, using the quadratic formula, or completing the square. Factoring is often the quickest method if possible.
We need two numbers that multiply to $-2$ and add to $-1$. These numbers are $-2$ and 1.

$$(x-2)(x+1) = 0$$

$$x-2 = 0 \implies x = 2$$ $$x+1 = 0 \implies x = -1$$

The critical values (x-coordinates of the intersection points) are $\boxed{x = 2 \text{ and } x = -1}$.