1/3g-4(2/3g-3)=2/3g-6

Solution for 1/3g-4(2/3g-3)=2/3g-6 equation:

1/3g-4(2/3g-3)=2/3g-6

We move all terms to the left:

1/3g-4(2/3g-3)-(2/3g-6)=0


Domain of the equation: 3g!=0

g!=0/3

g!=0

g∈R



Domain of the equation: 3g-3)!=0

g∈R



Domain of the equation: 3g-6)!=0

g∈R


We multiply parentheses

1/3g-8g-(2/3g-6)+12=0

We get rid of parentheses

1/3g-8g-2/3g+6+12=0

We multiply all the terms by the denominator


-8g*3g+6*3g+12*3g+1-2=0

We add all the numbers together, and all the variables

-8g*3g+6*3g+12*3g-1=0

Wy multiply elements

-24g^2+18g+36g-1=0

We add all the numbers together, and all the variables

-24g^2+54g-1=0

a = -24; b = 54; c = -1;
Δ = b2-4ac
Δ = 542-4·(-24)·(-1)
Δ = 2820
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:

$\sqrt{\Delta}=\sqrt{2820}=\sqrt{4*705}=\sqrt{4}*\sqrt{705}=2\sqrt{705}$
$g_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(54)-2\sqrt{705}}{2*-24}=\frac{-54-2\sqrt{705}}{-48}
$
$g_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(54)+2\sqrt{705}}{2*-24}=\frac{-54+2\sqrt{705}}{-48}
$