(3x-9)/(x-3)=9x

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

(3x-9)/(x-3)=9x

We move all terms to the left:

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


Domain of the equation: (x-3)!=0

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

x!=3

x∈R


We add all the numbers together, and all the variables

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

We multiply all the terms by the denominator


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

We multiply parentheses

-9x^2+27x+(3x-9)=0

We get rid of parentheses

-9x^2+27x+3x-9=0

We add all the numbers together, and all the variables

-9x^2+30x-9=0

a = -9; b = 30; c = -9;
Δ = b2-4ac
Δ = 302-4·(-9)·(-9)
Δ = 576
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{576}=24$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-24}{2*-9}=\frac{-54}{-18}
=+3
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+24}{2*-9}=\frac{-6}{-18}
=1/3
$