H(t)=-16t^2+56t+512

Solution for H(t)=-16t^2+56t+512 equation:

(H)=-16H^2+56H+512

We move all terms to the left:

(H)-(-16H^2+56H+512)=0

We get rid of parentheses

16H^2-56H+H-512=0

We add all the numbers together, and all the variables

16H^2-55H-512=0

a = 16; b = -55; c = -512;
Δ = b2-4ac
Δ = -552-4·16·(-512)
Δ = 35793
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:

$\sqrt{\Delta}=\sqrt{35793}=\sqrt{9*3977}=\sqrt{9}*\sqrt{3977}=3\sqrt{3977}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-55)-3\sqrt{3977}}{2*16}=\frac{55-3\sqrt{3977}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-55)+3\sqrt{3977}}{2*16}=\frac{55+3\sqrt{3977}}{32}
$