1/3x+2=8-6x

Solution for 1/3x+2=8-6x equation:

1/3x+2=8-6x

We move all terms to the left:

1/3x+2-(8-6x)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

1/3x-(-6x+8)+2=0

We get rid of parentheses

1/3x+6x-8+2=0

We multiply all the terms by the denominator


6x*3x-8*3x+2*3x+1=0

Wy multiply elements

18x^2-24x+6x+1=0

We add all the numbers together, and all the variables

18x^2-18x+1=0

a = 18; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·18·1
Δ = 252
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:

$\sqrt{\Delta}=\sqrt{252}=\sqrt{36*7}=\sqrt{36}*\sqrt{7}=6\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-6\sqrt{7}}{2*18}=\frac{18-6\sqrt{7}}{36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+6\sqrt{7}}{2*18}=\frac{18+6\sqrt{7}}{36}
$