S=250-5t-t^2

Solution for S=250-5t-t^2 equation:

=250-5S-S^2

We move all terms to the left:

-(250-5S-S^2)=0

We get rid of parentheses

S^2+5S-250=0

a = 1; b = 5; c = -250;
Δ = b2-4ac
Δ = 52-4·1·(-250)
Δ = 1025
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:

$\sqrt{\Delta}=\sqrt{1025}=\sqrt{25*41}=\sqrt{25}*\sqrt{41}=5\sqrt{41}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(5)-5\sqrt{41}}{2*1}=\frac{-5-5\sqrt{41}}{2}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+5\sqrt{41}}{2*1}=\frac{-5+5\sqrt{41}}{2}
$