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

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

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

We move all terms to the left:

-2x/x+2+x/3-(4/x+2)=0


Domain of the equation: x!=0

x∈R



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

x∈R


We get rid of parentheses

-2x/x+x/3-4/x-2+2=0

We calculate fractions


x^2/3x+(-2x-12)/3x-2+2=0

We add all the numbers together, and all the variables

x^2/3x+(-2x-12)/3x=0

We multiply all the terms by the denominator


x^2+(-2x-12)=0

We get rid of parentheses

x^2-2x-12=0

a = 1; b = -2; c = -12;
Δ = b2-4ac
Δ = -22-4·1·(-12)
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{-(-2)-2\sqrt{13}}{2*1}=\frac{2-2\sqrt{13}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{13}}{2*1}=\frac{2+2\sqrt{13}}{2}
$