H(t)=80+30t-5t2

Solution for H(t)=80+30t-5t2 equation:

(H)=80+30H-5H^2

We move all terms to the left:

(H)-(80+30H-5H^2)=0

We get rid of parentheses

5H^2-30H+H-80=0

We add all the numbers together, and all the variables

5H^2-29H-80=0

a = 5; b = -29; c = -80;
Δ = b2-4ac
Δ = -292-4·5·(-80)
Δ = 2441
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-29)-\sqrt{2441}}{2*5}=\frac{29-\sqrt{2441}}{10}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-29)+\sqrt{2441}}{2*5}=\frac{29+\sqrt{2441}}{10}
$