X-4/x+1+2x+1/x=3

Solution for X-4/x+1+2x+1/x=3 equation:

X-4/X+1+2X+1/X=3

We move all terms to the left:

X-4/X+1+2X+1/X-(3)=0


Domain of the equation: X!=0

X∈R


We add all the numbers together, and all the variables

3X-4/X+1/X-2=0

We multiply all the terms by the denominator


3X*X-2*X-4+1=0

We add all the numbers together, and all the variables

-2X+3X*X-3=0

Wy multiply elements

3X^2-2X-3=0

a = 3; b = -2; c = -3;
Δ = b2-4ac
Δ = -22-4·3·(-3)
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{10}}{2*3}=\frac{2-2\sqrt{10}}{6}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{10}}{2*3}=\frac{2+2\sqrt{10}}{6}
$