Quadratic function problem

Question

Find the coordinate of point A in the figure Can you solve it in way which doesn't involve derivation?

Answer 1

The parabola has zeros at $x=\pm 2$. Your blue line passes through $(-2,0)$ and $(0,-2)$ so its equation is $y=-x-2$. The blue line intersects the parabola a second time when $-(x+2) = -(x-2)(x+2)$ with $x\neq -2$, which occurs at $(3,-5)$. The black line is perpendicular to the blue one so it must have slope $1$. Therefore, its equation is $y = x-8$. So the point $A$, which is the $x-$intercept of the black line, must be $(8,0)$.

Answer 2

Another approach: The coordinates of intersections between the parabola and the straight line are $(-2,0)$ and $(3,-5)$. Now apply the geometric mean theorem on the triangle formed by those two points and $A$.