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

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

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

We move all terms to the left:

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

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

1/(x-1)+4/2x-5=0

We calculate fractions


2x/(2x^2-2x)+(4x-4)/(2x^2-2x)-5=0

We multiply all the terms by the denominator


2x+(4x-4)-5*(2x^2-2x)=0

We multiply parentheses

-10x^2+2x+(4x-4)+10x=0

We get rid of parentheses

-10x^2+2x+4x+10x-4=0

We add all the numbers together, and all the variables

-10x^2+16x-4=0

a = -10; b = 16; c = -4;
Δ = b2-4ac
Δ = 162-4·(-10)·(-4)
Δ = 96
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:

$\sqrt{\Delta}=\sqrt{96}=\sqrt{16*6}=\sqrt{16}*\sqrt{6}=4\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-4\sqrt{6}}{2*-10}=\frac{-16-4\sqrt{6}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+4\sqrt{6}}{2*-10}=\frac{-16+4\sqrt{6}}{-20}
$