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

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

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

We move all terms to the left:

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


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

x∈R



Domain of the equation: 2x-3)!=0

x∈R


We multiply parentheses

(2x-3/x-1)-4x+4x+12-3=0

We get rid of parentheses

2x-3/x-4x+4x-1+12-3=0

We multiply all the terms by the denominator


2x*x-4x*x+4x*x-1*x+12*x-3*x-3=0

We add all the numbers together, and all the variables

8x+2x*x-4x*x+4x*x-3=0

Wy multiply elements

2x^2-4x^2+4x^2+8x-3=0

We add all the numbers together, and all the variables

2x^2+8x-3=0

a = 2; b = 8; c = -3;
Δ = b2-4ac
Δ = 82-4·2·(-3)
Δ = 88
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:

$\sqrt{\Delta}=\sqrt{88}=\sqrt{4*22}=\sqrt{4}*\sqrt{22}=2\sqrt{22}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-2\sqrt{22}}{2*2}=\frac{-8-2\sqrt{22}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+2\sqrt{22}}{2*2}=\frac{-8+2\sqrt{22}}{4}
$