H=-5t^2+8t+3

Solution for H=-5t^2+8t+3 equation:

=-5H^2+8H+3

We move all terms to the left:

-(-5H^2+8H+3)=0

We get rid of parentheses

5H^2-8H-3=0

a = 5; b = -8; c = -3;
Δ = b2-4ac
Δ = -82-4·5·(-3)
Δ = 124
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{124}=\sqrt{4*31}=\sqrt{4}*\sqrt{31}=2\sqrt{31}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{31}}{2*5}=\frac{8-2\sqrt{31}}{10}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{31}}{2*5}=\frac{8+2\sqrt{31}}{10}
$