x-(1/3x-2)=12

Solution for x-(1/3x-2)=12 equation:

x-(1/3x-2)=12

We move all terms to the left:

x-(1/3x-2)-(12)=0


Domain of the equation: 3x-2)!=0

x∈R


We get rid of parentheses

x-1/3x+2-12=0

We multiply all the terms by the denominator


x*3x+2*3x-12*3x-1=0

Wy multiply elements

3x^2+6x-36x-1=0

We add all the numbers together, and all the variables

3x^2-30x-1=0

a = 3; b = -30; c = -1;
Δ = b2-4ac
Δ = -302-4·3·(-1)
Δ = 912
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:

$\sqrt{\Delta}=\sqrt{912}=\sqrt{16*57}=\sqrt{16}*\sqrt{57}=4\sqrt{57}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-4\sqrt{57}}{2*3}=\frac{30-4\sqrt{57}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+4\sqrt{57}}{2*3}=\frac{30+4\sqrt{57}}{6}
$