3x+1/x-1=5

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

3x+1/x-1=5

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

3x+1/x-6=0

We multiply all the terms by the denominator


3x*x-6*x+1=0

We add all the numbers together, and all the variables

-6x+3x*x+1=0

Wy multiply elements

3x^2-6x+1=0

a = 3; b = -6; c = +1;
Δ = b2-4ac
Δ = -62-4·3·1
Δ = 24
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:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{6}}{2*3}=\frac{6-2\sqrt{6}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{6}}{2*3}=\frac{6+2\sqrt{6}}{6}
$