5=x+1/x+1

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

5=x+1/x+1

We move all terms to the left:

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


Domain of the equation: x+1)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

4x-x*x-1=0

Wy multiply elements

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

a = -1; b = 4; c = -1;
Δ = b2-4ac
Δ = 42-4·(-1)·(-1)
Δ = 12
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:

$\sqrt{\Delta}=\sqrt{12}=\sqrt{4*3}=\sqrt{4}*\sqrt{3}=2\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{3}}{2*-1}=\frac{-4-2\sqrt{3}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{3}}{2*-1}=\frac{-4+2\sqrt{3}}{-2}
$