1/(9x-12)=3x+3

Solution for 1/(9x-12)=3x+3 equation:

1/(9x-12)=3x+3

We move all terms to the left:

1/(9x-12)-(3x+3)=0


Domain of the equation: (9x-12)!=0

We move all terms containing x to the left, all other terms to the right

9x!=12

x!=12/9

x!=1+1/3

x∈R


We get rid of parentheses

1/(9x-12)-3x-3=0

We multiply all the terms by the denominator


-3x*(9x-12)-3*(9x-12)+1=0

We multiply parentheses

-27x^2+36x-27x+36+1=0

We add all the numbers together, and all the variables

-27x^2+9x+37=0

a = -27; b = 9; c = +37;
Δ = b2-4ac
Δ = 92-4·(-27)·37
Δ = 4077
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:

$\sqrt{\Delta}=\sqrt{4077}=\sqrt{9*453}=\sqrt{9}*\sqrt{453}=3\sqrt{453}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(9)-3\sqrt{453}}{2*-27}=\frac{-9-3\sqrt{453}}{-54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(9)+3\sqrt{453}}{2*-27}=\frac{-9+3\sqrt{453}}{-54}
$