1/3x+2=5/x-2

Solution for 1/3x+2=5/x-2 equation:

1/3x+2=5/x-2

We move all terms to the left:

1/3x+2-(5/x-2)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



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

x∈R


We get rid of parentheses

1/3x-5/x+2+2=0

We calculate fractions


x/3x^2+(-15x)/3x^2+2+2=0

We add all the numbers together, and all the variables

x/3x^2+(-15x)/3x^2+4=0

We multiply all the terms by the denominator


x+(-15x)+4*3x^2=0

Wy multiply elements

12x^2+x+(-15x)=0

We get rid of parentheses

12x^2+x-15x=0

We add all the numbers together, and all the variables

12x^2-14x=0

a = 12; b = -14; c = 0;
Δ = b2-4ac
Δ = -142-4·12·0
Δ = 196
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{196}=14$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-14}{2*12}=\frac{0}{24}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+14}{2*12}=\frac{28}{24}
=1+1/6
$