H(t)=172-16t^2

Solution for H(t)=172-16t^2 equation:

(H)=172-16H^2

We move all terms to the left:

(H)-(172-16H^2)=0

We get rid of parentheses

16H^2+H-172=0

a = 16; b = 1; c = -172;
Δ = b2-4ac
Δ = 12-4·16·(-172)
Δ = 11009
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{11009}}{2*16}=\frac{-1-\sqrt{11009}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{11009}}{2*16}=\frac{-1+\sqrt{11009}}{32}
$