1/6x+2=5/6x-x-3

Solution for 1/6x+2=5/6x-x-3 equation:

1/6x+2=5/6x-x-3

We move all terms to the left:

1/6x+2-(5/6x-x-3)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R



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

x∈R


We add all the numbers together, and all the variables

1/6x-(-1x+5/6x-3)+2=0

We get rid of parentheses

1/6x+1x-5/6x+3+2=0

We multiply all the terms by the denominator


1x*6x+3*6x+2*6x+1-5=0

We add all the numbers together, and all the variables

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

Wy multiply elements

6x^2+18x+12x-4=0

We add all the numbers together, and all the variables

6x^2+30x-4=0

a = 6; b = 30; c = -4;
Δ = b2-4ac
Δ = 302-4·6·(-4)
Δ = 996
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:

$\sqrt{\Delta}=\sqrt{996}=\sqrt{4*249}=\sqrt{4}*\sqrt{249}=2\sqrt{249}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{249}}{2*6}=\frac{-30-2\sqrt{249}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{249}}{2*6}=\frac{-30+2\sqrt{249}}{12}
$