S=144+128t-16t^2

Solution for S=144+128t-16t^2 equation:

=144+128S-16S^2

We move all terms to the left:

-(144+128S-16S^2)=0

We get rid of parentheses

16S^2-128S-144=0

a = 16; b = -128; c = -144;
Δ = b2-4ac
Δ = -1282-4·16·(-144)
Δ = 25600
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{25600}=160$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-128)-160}{2*16}=\frac{-32}{32}
=-1
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-128)+160}{2*16}=\frac{288}{32}
=9
$