H(t)=-16t^2+72t+.5

Solution for H(t)=-16t^2+72t+.5 equation:

(H)=-16H^2+72H+.5

We move all terms to the left:

(H)-(-16H^2+72H+.5)=0

We get rid of parentheses

16H^2-72H+H-.5=0

We add all the numbers together, and all the variables

16H^2-71H-0.5=0

a = 16; b = -71; c = -0.5;
Δ = b2-4ac
Δ = -712-4·16·(-0.5)
Δ = 5073
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{-(-71)-\sqrt{5073}}{2*16}=\frac{71-\sqrt{5073}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-71)+\sqrt{5073}}{2*16}=\frac{71+\sqrt{5073}}{32}
$