x-1/6x=3x+2

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

x-1/6x=3x+2

We move all terms to the left:

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


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6x^2-18x^2-12x-1=0

We add all the numbers together, and all the variables

-12x^2-12x-1=0

a = -12; b = -12; c = -1;
Δ = b2-4ac
Δ = -122-4·(-12)·(-1)
Δ = 96
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:

$\sqrt{\Delta}=\sqrt{96}=\sqrt{16*6}=\sqrt{16}*\sqrt{6}=4\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-4\sqrt{6}}{2*-12}=\frac{12-4\sqrt{6}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+4\sqrt{6}}{2*-12}=\frac{12+4\sqrt{6}}{-24}
$