G(5x)=x+7/3x-2

Solution for G(5x)=x+7/3x-2 equation:

(5G)=G+7/3G-2

We move all terms to the left:

(5G)-(G+7/3G-2)=0


Domain of the equation: 3G-2)!=0

G∈R


We get rid of parentheses

5G-G-7/3G+2=0

We multiply all the terms by the denominator


5G*3G-G*3G+2*3G-7=0

Wy multiply elements

15G^2-3G^2+6G-7=0

We add all the numbers together, and all the variables

12G^2+6G-7=0

a = 12; b = 6; c = -7;
Δ = b2-4ac
Δ = 62-4·12·(-7)
Δ = 372
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:

$\sqrt{\Delta}=\sqrt{372}=\sqrt{4*93}=\sqrt{4}*\sqrt{93}=2\sqrt{93}$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{93}}{2*12}=\frac{-6-2\sqrt{93}}{24}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{93}}{2*12}=\frac{-6+2\sqrt{93}}{24}
$