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

Solution for 3x/(x-1)+2/x=4 equation:

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

We move all terms to the left:

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


Domain of the equation: (x-1)!=0

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

x!=1

x∈R



Domain of the equation: x!=0

x∈R


We calculate fractions


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

We multiply all the terms by the denominator


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

We multiply parentheses

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

We get rid of parentheses

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

We add all the numbers together, and all the variables

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

a = -1; b = 6; c = -2;
Δ = b2-4ac
Δ = 62-4·(-1)·(-2)
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{7}}{2*-1}=\frac{-6-2\sqrt{7}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{7}}{2*-1}=\frac{-6+2\sqrt{7}}{-2}
$