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

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

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

We move all terms to the left:

3x+1/x-1-(4/x-1+4)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: x-1+4)!=0

We move all terms containing x to the left, all other terms to the right

x+4)!=1

x∈R


We add all the numbers together, and all the variables

3x+1/x-(4/x+3)-1=0

We get rid of parentheses

3x+1/x-4/x-3-1=0

We multiply all the terms by the denominator


3x*x-3*x-1*x+1-4=0

We add all the numbers together, and all the variables

-4x+3x*x-3=0

Wy multiply elements

3x^2-4x-3=0

a = 3; b = -4; c = -3;
Δ = b2-4ac
Δ = -42-4·3·(-3)
Δ = 52
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:

$\sqrt{\Delta}=\sqrt{52}=\sqrt{4*13}=\sqrt{4}*\sqrt{13}=2\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{13}}{2*3}=\frac{4-2\sqrt{13}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{13}}{2*3}=\frac{4+2\sqrt{13}}{6}
$