G(x)=x2-x-5

Solution for G(x)=x2-x-5 equation:

(G)=G2-G-5

We move all terms to the left:

(G)-(G2-G-5)=0

We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We add all the numbers together, and all the variables

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

a = -1; b = 2; c = +5;
Δ = b2-4ac
Δ = 22-4·(-1)·5
Δ = 24
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(2)-2\sqrt{6}}{2*-1}=\frac{-2-2\sqrt{6}}{-2}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(2)+2\sqrt{6}}{2*-1}=\frac{-2+2\sqrt{6}}{-2}
$