F(-x)=x^2-8x+1

Solution for F(-x)=x^2-8x+1 equation:

(-F)=F^2-8F+1

We move all terms to the left:

(-F)-(F^2-8F+1)=0

We add all the numbers together, and all the variables

(-1F)-(F^2-8F+1)=0

We get rid of parentheses

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

We add all the numbers together, and all the variables

-1F^2+7F-1=0

a = -1; b = 7; c = -1;
Δ = b2-4ac
Δ = 72-4·(-1)·(-1)
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(7)-3\sqrt{5}}{2*-1}=\frac{-7-3\sqrt{5}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(7)+3\sqrt{5}}{2*-1}=\frac{-7+3\sqrt{5}}{-2}
$