5x+21=x+1/x+1

Solution for 5x+21=x+1/x+1 equation:

5x+21=x+1/x+1

We move all terms to the left:

5x+21-(x+1/x+1)=0


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

x∈R


We get rid of parentheses

5x-x-1/x-1+21=0

We multiply all the terms by the denominator


5x*x-x*x-1*x+21*x-1=0

We add all the numbers together, and all the variables

20x+5x*x-x*x-1=0

Wy multiply elements

5x^2-1x^2+20x-1=0

We add all the numbers together, and all the variables

4x^2+20x-1=0

a = 4; b = 20; c = -1;
Δ = b2-4ac
Δ = 202-4·4·(-1)
Δ = 416
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:

$\sqrt{\Delta}=\sqrt{416}=\sqrt{16*26}=\sqrt{16}*\sqrt{26}=4\sqrt{26}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-4\sqrt{26}}{2*4}=\frac{-20-4\sqrt{26}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+4\sqrt{26}}{2*4}=\frac{-20+4\sqrt{26}}{8}
$