90x-x^2=500

Solution for 90x-x^2=500 equation:

90x-x^2=500

We move all terms to the left:

90x-x^2-(500)=0

We add all the numbers together, and all the variables

-1x^2+90x-500=0

a = -1; b = 90; c = -500;
Δ = b2-4ac
Δ = 902-4·(-1)·(-500)
Δ = 6100
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6100}=\sqrt{100*61}=\sqrt{100}*\sqrt{61}=10\sqrt{61}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(90)-10\sqrt{61}}{2*-1}=\frac{-90-10\sqrt{61}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(90)+10\sqrt{61}}{2*-1}=\frac{-90+10\sqrt{61}}{-2}
$