2/3x+1/6=5/9x-2

Solution for 2/3x+1/6=5/9x-2 equation:

2/3x+1/6=5/9x-2

We move all terms to the left:

2/3x+1/6-(5/9x-2)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 9x-2)!=0

x∈R


We get rid of parentheses

2/3x-5/9x+2+1/6=0

We calculate fractions


243x^2/972x^2+648x/972x^2+(-540x)/972x^2+2=0

We multiply all the terms by the denominator


243x^2+648x+(-540x)+2*972x^2=0

Wy multiply elements

243x^2+1944x^2+648x+(-540x)=0

We get rid of parentheses

243x^2+1944x^2+648x-540x=0

We add all the numbers together, and all the variables

2187x^2+108x=0

a = 2187; b = 108; c = 0;
Δ = b2-4ac
Δ = 1082-4·2187·0
Δ = 11664
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{11664}=108$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(108)-108}{2*2187}=\frac{-216}{4374}
=-4/81
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(108)+108}{2*2187}=\frac{0}{4374}
=0
$