H=5+112t+-16t^2

Solution for H=5+112t+-16t^2 equation:

=5+112H+-16H^2

We move all terms to the left:

-(5+112H+-16H^2)=0

We use the square of the difference formula

-(5+112H-16H^2)=0

We get rid of parentheses

16H^2-112H-5=0

a = 16; b = -112; c = -5;
Δ = b2-4ac
Δ = -1122-4·16·(-5)
Δ = 12864
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:

$\sqrt{\Delta}=\sqrt{12864}=\sqrt{64*201}=\sqrt{64}*\sqrt{201}=8\sqrt{201}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-112)-8\sqrt{201}}{2*16}=\frac{112-8\sqrt{201}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-112)+8\sqrt{201}}{2*16}=\frac{112+8\sqrt{201}}{32}
$