H=5+64t+-16t2

Solution for H=5+64t+-16t2 equation:

=5+64H+-16H^2

We move all terms to the left:

-(5+64H+-16H^2)=0

We use the square of the difference formula

-(5+64H-16H^2)=0

We get rid of parentheses

16H^2-64H-5=0

a = 16; b = -64; c = -5;
Δ = b2-4ac
Δ = -642-4·16·(-5)
Δ = 4416
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:

$\sqrt{\Delta}=\sqrt{4416}=\sqrt{64*69}=\sqrt{64}*\sqrt{69}=8\sqrt{69}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-64)-8\sqrt{69}}{2*16}=\frac{64-8\sqrt{69}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-64)+8\sqrt{69}}{2*16}=\frac{64+8\sqrt{69}}{32}
$