3x=5/9x+2

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

3x=5/9x+2

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

3x-5/9x-2=0

We multiply all the terms by the denominator


3x*9x-2*9x-5=0

Wy multiply elements

27x^2-18x-5=0

a = 27; b = -18; c = -5;
Δ = b2-4ac
Δ = -182-4·27·(-5)
Δ = 864
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{864}=\sqrt{144*6}=\sqrt{144}*\sqrt{6}=12\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-12\sqrt{6}}{2*27}=\frac{18-12\sqrt{6}}{54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+12\sqrt{6}}{2*27}=\frac{18+12\sqrt{6}}{54}
$