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

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



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

We move all terms to the left:

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



Domain of the equation: x!=0

x∈R





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

We calculate fractions


(-15x+1)/3x^2+2x/3x^2+2=0

We multiply all the terms by the denominator


(-15x+1)+2x+2*3x^2=0

We add all the numbers together, and all the variables

2x+(-15x+1)+2*3x^2=0

Wy multiply elements

6x^2+2x+(-15x+1)=0

We get rid of parentheses

6x^2+2x-15x+1=0

We add all the numbers together, and all the variables

6x^2-13x+1=0

a = 6; b = -13; c = +1;
Δ = b2-4ac
Δ = -132-4·6·1
Δ = 145
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-13)-\sqrt{145}}{2*6}=\frac{13-\sqrt{145}}{12}
$


$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-13)+\sqrt{145}}{2*6}=\frac{13+\sqrt{145}}{12}
$

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