(3x+6/3)-4x/(x-3)=x

Solution for (3x+6/3)-4x/(x-3)=x equation:

(3x+6/3)-4x/(x-3)=x

We move all terms to the left:

(3x+6/3)-4x/(x-3)-(x)=0


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

We move all terms containing x to the left, all other terms to the right

x!=3

x∈R


We add all the numbers together, and all the variables

(3x+2)-4x/(x-3)-x=0

We add all the numbers together, and all the variables

-1x+(3x+2)-4x/(x-3)=0

We get rid of parentheses

-1x+3x-4x/(x-3)+2=0

We multiply all the terms by the denominator


-1x*(x-3)+3x*(x-3)-4x+2*(x-3)=0

We add all the numbers together, and all the variables

-4x-1x*(x-3)+3x*(x-3)+2*(x-3)=0

We multiply parentheses

-x^2+3x^2-4x+3x-9x+2x-6=0

We add all the numbers together, and all the variables

2x^2-8x-6=0

a = 2; b = -8; c = -6;
Δ = b2-4ac
Δ = -82-4·2·(-6)
Δ = 112
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:

$\sqrt{\Delta}=\sqrt{112}=\sqrt{16*7}=\sqrt{16}*\sqrt{7}=4\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{7}}{2*2}=\frac{8-4\sqrt{7}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{7}}{2*2}=\frac{8+4\sqrt{7}}{4}
$