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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 3x-2)!=0

x∈R


We get rid of parentheses

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

We calculate fractions


27x/54x^2+(-2x)/54x^2+2x/54x^2+2=0

We multiply all the terms by the denominator


27x+(-2x)+2x+2*54x^2=0

We add all the numbers together, and all the variables

29x+(-2x)+2*54x^2=0

Wy multiply elements

108x^2+29x+(-2x)=0

We get rid of parentheses

108x^2+29x-2x=0

We add all the numbers together, and all the variables

108x^2+27x=0

a = 108; b = 27; c = 0;
Δ = b2-4ac
Δ = 272-4·108·0
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{729}=27$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(27)-27}{2*108}=\frac{-54}{216}
=-1/4
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(27)+27}{2*108}=\frac{0}{216}
=0
$