2+1/6x=3x+4

Solution for 2+1/6x=3x+4 equation:

2+1/6x=3x+4

We move all terms to the left:

2+1/6x-(3x+4)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R


We get rid of parentheses

1/6x-3x-4+2=0

We multiply all the terms by the denominator


-3x*6x-4*6x+2*6x+1=0

Wy multiply elements

-18x^2-24x+12x+1=0

We add all the numbers together, and all the variables

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

a = -18; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·(-18)·1
Δ = 216
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:

$\sqrt{\Delta}=\sqrt{216}=\sqrt{36*6}=\sqrt{36}*\sqrt{6}=6\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-6\sqrt{6}}{2*-18}=\frac{12-6\sqrt{6}}{-36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+6\sqrt{6}}{2*-18}=\frac{12+6\sqrt{6}}{-36}
$