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

-4x+2x*x-x*x-1=0

Wy multiply elements

2x^2-1x^2-4x-1=0

We add all the numbers together, and all the variables

x^2-4x-1=0

a = 1; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·1·(-1)
Δ = 20
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:

$\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{5}}{2*1}=\frac{4-2\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{5}}{2*1}=\frac{4+2\sqrt{5}}{2}
$