S=-16t^2+51+69

Solution for S=-16t^2+51+69 equation:

=-16S^2+51+69

We move all terms to the left:

-(-16S^2+51+69)=0

We get rid of parentheses

16S^2-51-69=0

We add all the numbers together, and all the variables

16S^2-120=0

a = 16; b = 0; c = -120;
Δ = b2-4ac
Δ = 02-4·16·(-120)
Δ = 7680
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:

$\sqrt{\Delta}=\sqrt{7680}=\sqrt{256*30}=\sqrt{256}*\sqrt{30}=16\sqrt{30}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-16\sqrt{30}}{2*16}=\frac{0-16\sqrt{30}}{32}
=-\frac{16\sqrt{30}}{32}
=-\frac{\sqrt{30}}{2}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+16\sqrt{30}}{2*16}=\frac{0+16\sqrt{30}}{32}
=\frac{16\sqrt{30}}{32}
=\frac{\sqrt{30}}{2}
$