(a) Determine all real numbers $x$ for which $x^{2}+2 x-8=0$.

(b) Determine the values of $b$ and $c$ for which the parabola with equation $y=x^{2}+b x+c$ passes through the points $(1,2)$ and $(2,0)$.

(c) A ball is thrown from a window at the point $(0,2)$. The ball flies through the air along the parabola with equation $y=a(x-1)^{2}+\frac{8}{3}$, for some real number $a$. Determine the positive real number $d$ for which the ball hits the ground at $(d, 0)$.

![](https://cdn.mathpix.com/cropped/2024_04_17_ff0a25f37b4b6ded9838g-3.jpg?height=517&width=572&top_left_y=1861&top_left_x=1161)