G(1/2)=-x^2+4x+1

Solution for G(1/2)=-x^2+4x+1 equation:

(1/2)=-G^2+4G+1

We move all terms to the left:

(1/2)-(-G^2+4G+1)=0

We add all the numbers together, and all the variables

-(-G^2+4G+1)+(+1/2)=0

We get rid of parentheses

G^2-4G-1+1/2=0

We multiply all the terms by the denominator


G^2*2-4G*2+1-1*2=0

We add all the numbers together, and all the variables

G^2*2-4G*2-1=0

Wy multiply elements

2G^2-8G-1=0

a = 2; b = -8; c = -1;
Δ = b2-4ac
Δ = -82-4·2·(-1)
Δ = 72
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:

$\sqrt{\Delta}=\sqrt{72}=\sqrt{36*2}=\sqrt{36}*\sqrt{2}=6\sqrt{2}$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-6\sqrt{2}}{2*2}=\frac{8-6\sqrt{2}}{4}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+6\sqrt{2}}{2*2}=\frac{8+6\sqrt{2}}{4}
$