S(t)=40t-409t^2

Solution for S(t)=40t-409t^2 equation:

(S)=40S-409S^2

We move all terms to the left:

(S)-(40S-409S^2)=0

We get rid of parentheses

409S^2-40S+S=0

We add all the numbers together, and all the variables

409S^2-39S=0

a = 409; b = -39; c = 0;
Δ = b2-4ac
Δ = -392-4·409·0
Δ = 1521
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1521}=39$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-39)-39}{2*409}=\frac{0}{818}
=0
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-39)+39}{2*409}=\frac{78}{818}
=39/409
$