2/3x=x-6

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

2/3x=x-6

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

2/3x-x+6=0

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -3; b = 18; c = +2;
Δ = b2-4ac
Δ = 182-4·(-3)·2
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{87}}{2*-3}=\frac{-18-2\sqrt{87}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{87}}{2*-3}=\frac{-18+2\sqrt{87}}{-6}
$