H(t)=500t-5t^2

Solution for H(t)=500t-5t^2 equation:

(H)=500H-5H^2

We move all terms to the left:

(H)-(500H-5H^2)=0

We get rid of parentheses

5H^2-500H+H=0

We add all the numbers together, and all the variables

5H^2-499H=0

a = 5; b = -499; c = 0;
Δ = b2-4ac
Δ = -4992-4·5·0
Δ = 249001
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{249001}=499$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-499)-499}{2*5}=\frac{0}{10}
=0
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-499)+499}{2*5}=\frac{998}{10}
=99+4/5
$