F(x+8)=x^2+9x-9

Solution for F(x+8)=x^2+9x-9 equation:

(F+8)=F^2+9F-9

We move all terms to the left:

(F+8)-(F^2+9F-9)=0

We get rid of parentheses

-F^2+F-9F+8+9=0

We add all the numbers together, and all the variables

-1F^2-8F+17=0

a = -1; b = -8; c = +17;
Δ = b2-4ac
Δ = -82-4·(-1)·17
Δ = 132
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{132}=\sqrt{4*33}=\sqrt{4}*\sqrt{33}=2\sqrt{33}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{33}}{2*-1}=\frac{8-2\sqrt{33}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{33}}{2*-1}=\frac{8+2\sqrt{33}}{-2}
$