S(t)=-16t^2+800

Solution for S(t)=-16t^2+800 equation:

(S)=-16S^2+800

We move all terms to the left:

(S)-(-16S^2+800)=0

We get rid of parentheses

16S^2+S-800=0

a = 16; b = 1; c = -800;
Δ = b2-4ac
Δ = 12-4·16·(-800)
Δ = 51201
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{51201}=\sqrt{9*5689}=\sqrt{9}*\sqrt{5689}=3\sqrt{5689}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-3\sqrt{5689}}{2*16}=\frac{-1-3\sqrt{5689}}{32}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+3\sqrt{5689}}{2*16}=\frac{-1+3\sqrt{5689}}{32}
$