H(1/2)=-16t^2+16t+32

Solution for H(1/2)=-16t^2+16t+32 equation:

(1/2)=-16H^2+16H+32

We move all terms to the left:

(1/2)-(-16H^2+16H+32)=0

We add all the numbers together, and all the variables

-(-16H^2+16H+32)+(+1/2)=0

We get rid of parentheses

16H^2-16H-32+1/2=0

We multiply all the terms by the denominator


16H^2*2-16H*2+1-32*2=0

We add all the numbers together, and all the variables

16H^2*2-16H*2-63=0

Wy multiply elements

32H^2-32H-63=0

a = 32; b = -32; c = -63;
Δ = b2-4ac
Δ = -322-4·32·(-63)
Δ = 9088
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:

$\sqrt{\Delta}=\sqrt{9088}=\sqrt{64*142}=\sqrt{64}*\sqrt{142}=8\sqrt{142}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-32)-8\sqrt{142}}{2*32}=\frac{32-8\sqrt{142}}{64}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-32)+8\sqrt{142}}{2*32}=\frac{32+8\sqrt{142}}{64}
$