1/3x+6=x-2

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

1/3x+6=x-2

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-3x^2+6x+18x+1=0

We add all the numbers together, and all the variables

-3x^2+24x+1=0

a = -3; b = 24; c = +1;
Δ = b2-4ac
Δ = 242-4·(-3)·1
Δ = 588
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:

$\sqrt{\Delta}=\sqrt{588}=\sqrt{196*3}=\sqrt{196}*\sqrt{3}=14\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-14\sqrt{3}}{2*-3}=\frac{-24-14\sqrt{3}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+14\sqrt{3}}{2*-3}=\frac{-24+14\sqrt{3}}{-6}
$