6x-1/3x-2=3x+2

Solution for 6x-1/3x-2=3x+2 equation:

6x-1/3x-2=3x+2

We move all terms to the left:

6x-1/3x-2-(3x+2)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

6x-1/3x-3x-2-2=0

We multiply all the terms by the denominator


6x*3x-3x*3x-2*3x-2*3x-1=0

Wy multiply elements

18x^2-9x^2-6x-6x-1=0

We add all the numbers together, and all the variables

9x^2-12x-1=0

a = 9; b = -12; c = -1;
Δ = b2-4ac
Δ = -122-4·9·(-1)
Δ = 180
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{180}=\sqrt{36*5}=\sqrt{36}*\sqrt{5}=6\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-6\sqrt{5}}{2*9}=\frac{12-6\sqrt{5}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+6\sqrt{5}}{2*9}=\frac{12+6\sqrt{5}}{18}
$