S=5t^2+17t+12

Solution for S=5t^2+17t+12 equation:

=5S^2+17S+12

We move all terms to the left:

-(5S^2+17S+12)=0

We get rid of parentheses

-5S^2-17S-12=0

a = -5; b = -17; c = -12;
Δ = b2-4ac
Δ = -172-4·(-5)·(-12)
Δ = 49
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{49}=7$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-17)-7}{2*-5}=\frac{10}{-10}
=-1
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-17)+7}{2*-5}=\frac{24}{-10}
=-2+2/5
$