G(-1/3)=x-2x^2

Solution for G(-1/3)=x-2x^2 equation:

(-1/3)=G-2G^2

We move all terms to the left:

(-1/3)-(G-2G^2)=0

We get rid of parentheses

2G^2-G-1/3=0

We multiply all the terms by the denominator


2G^2*3-G*3-1=0

Wy multiply elements

6G^2-3G-1=0

a = 6; b = -3; c = -1;
Δ = b2-4ac
Δ = -32-4·6·(-1)
Δ = 33
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-\sqrt{33}}{2*6}=\frac{3-\sqrt{33}}{12}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+\sqrt{33}}{2*6}=\frac{3+\sqrt{33}}{12}
$