X+1/x=2x+1

Solution for X+1/x=2x+1 equation:

X+1/X=2X+1

We move all terms to the left:

X+1/X-(2X+1)=0


Domain of the equation: X!=0

X∈R


We get rid of parentheses

X+1/X-2X-1=0

We multiply all the terms by the denominator


X*X-2X*X-1*X+1=0

We add all the numbers together, and all the variables

-1X+X*X-2X*X+1=0

Wy multiply elements

X^2-2X^2-1X+1=0

We add all the numbers together, and all the variables

-1X^2-1X+1=0

a = -1; b = -1; c = +1;
Δ = b2-4ac
Δ = -12-4·(-1)·1
Δ = 5
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

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