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

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

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

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x-5)!=0

x∈R


We get rid of parentheses

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

We calculate fractions


x/3x^2+(-6x)/3x^2+5-2=0

We add all the numbers together, and all the variables

x/3x^2+(-6x)/3x^2+3=0

We multiply all the terms by the denominator


x+(-6x)+3*3x^2=0

Wy multiply elements

9x^2+x+(-6x)=0

We get rid of parentheses

9x^2+x-6x=0

We add all the numbers together, and all the variables

9x^2-5x=0

a = 9; b = -5; c = 0;
Δ = b2-4ac
Δ = -52-4·9·0
Δ = 25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{25}=5$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-5)-5}{2*9}=\frac{0}{18}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+5}{2*9}=\frac{10}{18}
=5/9
$