F(x+4)=x^2+7x-7

Solution for F(x+4)=x^2+7x-7 equation:

(F+4)=F^2+7F-7

We move all terms to the left:

(F+4)-(F^2+7F-7)=0

We get rid of parentheses

-F^2+F-7F+4+7=0

We add all the numbers together, and all the variables

-1F^2-6F+11=0

a = -1; b = -6; c = +11;
Δ = b2-4ac
Δ = -62-4·(-1)·11
Δ = 80
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:

$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-4\sqrt{5}}{2*-1}=\frac{6-4\sqrt{5}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+4\sqrt{5}}{2*-1}=\frac{6+4\sqrt{5}}{-2}
$