2(x-5)-2/x-1=x-1

Solution for 2(x-5)-2/x-1=x-1 equation:

2(x-5)-2/x-1=x-1

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R


We multiply parentheses

2x-2/x-(x-1)-10-1=0

We get rid of parentheses

2x-2/x-x+1-10-1=0

We multiply all the terms by the denominator


2x*x-x*x+1*x-10*x-1*x-2=0

We add all the numbers together, and all the variables

-10x+2x*x-x*x-2=0

Wy multiply elements

2x^2-1x^2-10x-2=0

We add all the numbers together, and all the variables

x^2-10x-2=0

a = 1; b = -10; c = -2;
Δ = b2-4ac
Δ = -102-4·1·(-2)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-6\sqrt{3}}{2*1}=\frac{10-6\sqrt{3}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+6\sqrt{3}}{2*1}=\frac{10+6\sqrt{3}}{2}
$