F(x)=x2-5x+1

Solution for F(x)=x2-5x+1 equation:

(F)=F2-5F+1

We move all terms to the left:

(F)-(F2-5F+1)=0

We add all the numbers together, and all the variables

-(+F^2-5F+1)+F=0

We get rid of parentheses

-F^2+5F+F-1=0

We add all the numbers together, and all the variables

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

a = -1; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·(-1)·(-1)
Δ = 32
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:

$\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=\sqrt{16}*\sqrt{2}=4\sqrt{2}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-4\sqrt{2}}{2*-1}=\frac{-6-4\sqrt{2}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+4\sqrt{2}}{2*-1}=\frac{-6+4\sqrt{2}}{-2}
$