S(t)=3t^2-14t-24

Solution for S(t)=3t^2-14t-24 equation:

(S)=3S^2-14S-24

We move all terms to the left:

(S)-(3S^2-14S-24)=0

We get rid of parentheses

-3S^2+S+14S+24=0

We add all the numbers together, and all the variables

-3S^2+15S+24=0

a = -3; b = 15; c = +24;
Δ = b2-4ac
Δ = 152-4·(-3)·24
Δ = 513
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:

$\sqrt{\Delta}=\sqrt{513}=\sqrt{9*57}=\sqrt{9}*\sqrt{57}=3\sqrt{57}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(15)-3\sqrt{57}}{2*-3}=\frac{-15-3\sqrt{57}}{-6}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(15)+3\sqrt{57}}{2*-3}=\frac{-15+3\sqrt{57}}{-6}
$