(4x-9)/(x-1)=2/2x-2

Solution for (4x-9)/(x-1)=2/2x-2 equation:

(4x-9)/(x-1)=2/2x-2

We move all terms to the left:

(4x-9)/(x-1)-(2/2x-2)=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-2)!=0

x∈R


We get rid of parentheses

(4x-9)/(x-1)-2/2x+2=0

We calculate fractions


(8x^2-18x)/(2x^2-2x)+(-2x+2)/(2x^2-2x)+2=0

We multiply all the terms by the denominator


(8x^2-18x)+(-2x+2)+2*(2x^2-2x)=0

We multiply parentheses

4x^2+(8x^2-18x)+(-2x+2)-4x=0

We get rid of parentheses

4x^2+8x^2-18x-2x-4x+2=0

We add all the numbers together, and all the variables

12x^2-24x+2=0

a = 12; b = -24; c = +2;
Δ = b2-4ac
Δ = -242-4·12·2
Δ = 480
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:

$\sqrt{\Delta}=\sqrt{480}=\sqrt{16*30}=\sqrt{16}*\sqrt{30}=4\sqrt{30}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-4\sqrt{30}}{2*12}=\frac{24-4\sqrt{30}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+4\sqrt{30}}{2*12}=\frac{24+4\sqrt{30}}{24}
$