3x-5=1/x-1

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

3x-5=1/x-1

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

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

Wy multiply elements

3x^2-4x-1=0

a = 3; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·3·(-1)
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{7}}{2*3}=\frac{4-2\sqrt{7}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{7}}{2*3}=\frac{4+2\sqrt{7}}{6}
$