S(t)=30+50t-9.8t^2

Solution for S(t)=30+50t-9.8t^2 equation:

(S)=30+50S-9.8S^2

We move all terms to the left:

(S)-(30+50S-9.8S^2)=0

We get rid of parentheses

9.8S^2-50S+S-30=0

We add all the numbers together, and all the variables

9.8S^2-49S-30=0

a = 9.8; b = -49; c = -30;
Δ = b2-4ac
Δ = -492-4·9.8·(-30)
Δ = 3577
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:

$\sqrt{\Delta}=\sqrt{3577}=\sqrt{49*73}=\sqrt{49}*\sqrt{73}=7\sqrt{73}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-49)-7\sqrt{73}}{2*9.8}=\frac{49-7\sqrt{73}}{19.6}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-49)+7\sqrt{73}}{2*9.8}=\frac{49+7\sqrt{73}}{19.6}
$