G(x)=1/x-1

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

(G)=1/G-1

We move all terms to the left:

(G)-(1/G-1)=0


Domain of the equation: G-1)!=0

G∈R


We get rid of parentheses

G-1/G+1=0

We multiply all the terms by the denominator


G*G+1*G-1=0

We add all the numbers together, and all the variables

G+G*G-1=0

Wy multiply elements

G^2+G-1=0

a = 1; b = 1; c = -1;
Δ = b2-4ac
Δ = 12-4·1·(-1)
Δ = 5
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{-(1)-\sqrt{5}}{2*1}=\frac{-1-\sqrt{5}}{2}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{5}}{2*1}=\frac{-1+\sqrt{5}}{2}
$