x=x^2-90

Solution for x=x^2-90 equation:

x=x^2-90

We move all terms to the left:

x-(x^2-90)=0

We get rid of parentheses

-x^2+x+90=0

We add all the numbers together, and all the variables

-1x^2+x+90=0

a = -1; b = 1; c = +90;
Δ = b2-4ac
Δ = 12-4·(-1)·90
Δ = 361
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{361}=19$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-19}{2*-1}=\frac{-20}{-2}
=+10
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+19}{2*-1}=\frac{18}{-2}
=-9
$