G(6)x=17-2/3x

Solution for G(6)x=17-2/3x equation:

(6)G=17-2/3G

We move all terms to the left:

(6)G-(17-2/3G)=0


Domain of the equation: 3G)!=0

G!=0/1

G!=0

G∈R


We add all the numbers together, and all the variables

6G-(-2/3G+17)=0

We get rid of parentheses

6G+2/3G-17=0

We multiply all the terms by the denominator


6G*3G-17*3G+2=0

Wy multiply elements

18G^2-51G+2=0

a = 18; b = -51; c = +2;
Δ = b2-4ac
Δ = -512-4·18·2
Δ = 2457
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:

$\sqrt{\Delta}=\sqrt{2457}=\sqrt{9*273}=\sqrt{9}*\sqrt{273}=3\sqrt{273}$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-51)-3\sqrt{273}}{2*18}=\frac{51-3\sqrt{273}}{36}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-51)+3\sqrt{273}}{2*18}=\frac{51+3\sqrt{273}}{36}
$