S(t)=-16t^2+144t+6

Solution for S(t)=-16t^2+144t+6 equation:

(S)=-16S^2+144S+6

We move all terms to the left:

(S)-(-16S^2+144S+6)=0

We get rid of parentheses

16S^2-144S+S-6=0

We add all the numbers together, and all the variables

16S^2-143S-6=0

a = 16; b = -143; c = -6;
Δ = b2-4ac
Δ = -1432-4·16·(-6)
Δ = 20833
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-143)-\sqrt{20833}}{2*16}=\frac{143-\sqrt{20833}}{32}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-143)+\sqrt{20833}}{2*16}=\frac{143+\sqrt{20833}}{32}
$