H=-16t^2+144t+288

Solution for H=-16t^2+144t+288 equation:

=-16H^2+144H+288

We move all terms to the left:

-(-16H^2+144H+288)=0

We get rid of parentheses

16H^2-144H-288=0

a = 16; b = -144; c = -288;
Δ = b2-4ac
Δ = -1442-4·16·(-288)
Δ = 39168
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:

$\sqrt{\Delta}=\sqrt{39168}=\sqrt{2304*17}=\sqrt{2304}*\sqrt{17}=48\sqrt{17}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-144)-48\sqrt{17}}{2*16}=\frac{144-48\sqrt{17}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-144)+48\sqrt{17}}{2*16}=\frac{144+48\sqrt{17}}{32}
$