H(t)=-16t^2+85t+1

Solution for H(t)=-16t^2+85t+1 equation:

(H)=-16H^2+85H+1

We move all terms to the left:

(H)-(-16H^2+85H+1)=0

We get rid of parentheses

16H^2-85H+H-1=0

We add all the numbers together, and all the variables

16H^2-84H-1=0

a = 16; b = -84; c = -1;
Δ = b2-4ac
Δ = -842-4·16·(-1)
Δ = 7120
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:

$\sqrt{\Delta}=\sqrt{7120}=\sqrt{16*445}=\sqrt{16}*\sqrt{445}=4\sqrt{445}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-84)-4\sqrt{445}}{2*16}=\frac{84-4\sqrt{445}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-84)+4\sqrt{445}}{2*16}=\frac{84+4\sqrt{445}}{32}
$