H(t)=(16t^2)+75t+80

Solution for H(t)=(16t^2)+75t+80 equation:

(H)=(16H^2)+75H+80

We move all terms to the left:

(H)-((16H^2)+75H+80)=0

We get rid of parentheses

-16H^2+H-75H-80=0

We add all the numbers together, and all the variables

-16H^2-74H-80=0

a = -16; b = -74; c = -80;
Δ = b2-4ac
Δ = -742-4·(-16)·(-80)
Δ = 356
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:

$\sqrt{\Delta}=\sqrt{356}=\sqrt{4*89}=\sqrt{4}*\sqrt{89}=2\sqrt{89}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-74)-2\sqrt{89}}{2*-16}=\frac{74-2\sqrt{89}}{-32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-74)+2\sqrt{89}}{2*-16}=\frac{74+2\sqrt{89}}{-32}
$