x=1/x+2

Solution for x=1/x+2 equation:

x=1/x+2

We move all terms to the left:

x-(1/x+2)=0


Domain of the equation: x+2)!=0

x∈R


We get rid of parentheses

x-1/x-2=0

We multiply all the terms by the denominator


x*x-2*x-1=0

We add all the numbers together, and all the variables

-2x+x*x-1=0

Wy multiply elements

x^2-2x-1=0

a = 1; b = -2; c = -1;
Δ = b2-4ac
Δ = -22-4·1·(-1)
Δ = 8
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:

$\sqrt{\Delta}=\sqrt{8}=\sqrt{4*2}=\sqrt{4}*\sqrt{2}=2\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{2}}{2*1}=\frac{2-2\sqrt{2}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{2}}{2*1}=\frac{2+2\sqrt{2}}{2}
$