1/3x-6=2x-27

Solution for 1/3x-6=2x-27 equation:

1/3x-6=2x-27

We move all terms to the left:

1/3x-6-(2x-27)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x-2x+27-6=0

We multiply all the terms by the denominator


-2x*3x+27*3x-6*3x+1=0

Wy multiply elements

-6x^2+81x-18x+1=0

We add all the numbers together, and all the variables

-6x^2+63x+1=0

a = -6; b = 63; c = +1;
Δ = b2-4ac
Δ = 632-4·(-6)·1
Δ = 3993
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:

$\sqrt{\Delta}=\sqrt{3993}=\sqrt{121*33}=\sqrt{121}*\sqrt{33}=11\sqrt{33}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(63)-11\sqrt{33}}{2*-6}=\frac{-63-11\sqrt{33}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(63)+11\sqrt{33}}{2*-6}=\frac{-63+11\sqrt{33}}{-12}
$