18x/3x+1=1/x-2

Solution for 18x/3x+1=1/x-2 equation:

18x/3x+1=1/x-2

We move all terms to the left:

18x/3x+1-(1/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

18x/3x-1/x+2+1=0

We calculate fractions


18x^2/3x^2+(-3x)/3x^2+2+1=0

We add all the numbers together, and all the variables

18x^2/3x^2+(-3x)/3x^2+3=0

We multiply all the terms by the denominator


18x^2+(-3x)+3*3x^2=0

Wy multiply elements

18x^2+9x^2+(-3x)=0

We get rid of parentheses

18x^2+9x^2-3x=0

We add all the numbers together, and all the variables

27x^2-3x=0

a = 27; b = -3; c = 0;
Δ = b2-4ac
Δ = -32-4·27·0
Δ = 9
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{9}=3$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-3}{2*27}=\frac{0}{54}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+3}{2*27}=\frac{6}{54}
=1/9
$