h(.5h)=5000

Solution for h(.5h)=5000 equation:

h(.5h)=5000

We move all terms to the left:

h(.5h)-(5000)=0

We add all the numbers together, and all the variables

h(+.5h)-5000=0

We multiply parentheses

h^2-5000=0

a = 1; b = 0; c = -5000;
Δ = b2-4ac
Δ = 02-4·1·(-5000)
Δ = 20000
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:

$\sqrt{\Delta}=\sqrt{20000}=\sqrt{10000*2}=\sqrt{10000}*\sqrt{2}=100\sqrt{2}$
$h_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-100\sqrt{2}}{2*1}=\frac{0-100\sqrt{2}}{2}
=-\frac{100\sqrt{2}}{2}
=-50\sqrt{2}
$
$h_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+100\sqrt{2}}{2*1}=\frac{0+100\sqrt{2}}{2}
=\frac{100\sqrt{2}}{2}
=50\sqrt{2}
$