1/9x+6=3x-2

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

1/9x+6=3x-2

We move all terms to the left:

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


Domain of the equation: 9x!=0

x!=0/9

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


-3x*9x+2*9x+6*9x+1=0

Wy multiply elements

-27x^2+18x+54x+1=0

We add all the numbers together, and all the variables

-27x^2+72x+1=0

a = -27; b = 72; c = +1;
Δ = b2-4ac
Δ = 722-4·(-27)·1
Δ = 5292
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:

$\sqrt{\Delta}=\sqrt{5292}=\sqrt{1764*3}=\sqrt{1764}*\sqrt{3}=42\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(72)-42\sqrt{3}}{2*-27}=\frac{-72-42\sqrt{3}}{-54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(72)+42\sqrt{3}}{2*-27}=\frac{-72+42\sqrt{3}}{-54}
$