Answer:
To find the points of intersection, we can set the equations of the graphs equal to each other:
$x^2 = -x$
Rearranging the terms, we have:
$x^2 + x = 0$
Factoring out an $x$, we get:
$x(x + 1) = 0$
This equation is satisfied when $x = 0$ or $x = -1$.
Substituting these values back into the equation $y = x^2$, we find the corresponding $y$-coordinates.
For $x = 0$, we have $y = (0)^2 = 0$.
For $x = -1$, we have $y = (-1)^2 = 1$.
Therefore, the two points of intersection are $(0, 0)$ and $(-1, 1)$.
To find the distance between these two points, we can use the distance formula:
$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Using the coordinates of the two points, we have:
$\text{Distance} = \sqrt{((-1) - 0)^2 + (1 - 0)^2}$
Simplifying,
$\text{Distance} = \sqrt{1 + 1}$
$\text{Distance} = \sqrt{2}$
Therefore, the distance between the two points of intersection is $\sqrt{2}$.