2x+1=x+1/x-5

Solution for 2x+1=x+1/x-5 equation:

2x+1=x+1/x-5

We move all terms to the left:

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


Domain of the equation: x-5)!=0

x∈R


We get rid of parentheses

2x-x-1/x+5+1=0

We multiply all the terms by the denominator


2x*x-x*x+5*x+1*x-1=0

We add all the numbers together, and all the variables

6x+2x*x-x*x-1=0

Wy multiply elements

2x^2-1x^2+6x-1=0

We add all the numbers together, and all the variables

x^2+6x-1=0

a = 1; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·1·(-1)
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{10}}{2*1}=\frac{-6-2\sqrt{10}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{10}}{2*1}=\frac{-6+2\sqrt{10}}{2}
$