2x-3=4x-5/x-1=x-1

Solution for 2x-3=4x-5/x-1=x-1 equation:

2x-3=4x-5/x-1=x-1

We move all terms to the left:

2x-3-(4x-5/x-1)=0


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

x∈R


We get rid of parentheses

2x-4x+5/x+1-3=0

We multiply all the terms by the denominator


2x*x-4x*x+1*x-3*x+5=0

We add all the numbers together, and all the variables

-2x+2x*x-4x*x+5=0

Wy multiply elements

2x^2-4x^2-2x+5=0

We add all the numbers together, and all the variables

-2x^2-2x+5=0

a = -2; b = -2; c = +5;
Δ = b2-4ac
Δ = -22-4·(-2)·5
Δ = 44
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:

$\sqrt{\Delta}=\sqrt{44}=\sqrt{4*11}=\sqrt{4}*\sqrt{11}=2\sqrt{11}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{11}}{2*-2}=\frac{2-2\sqrt{11}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{11}}{2*-2}=\frac{2+2\sqrt{11}}{-4}
$