(-4x+18x+3)/(6x-9)=x

Solution for (-4x+18x+3)/(6x-9)=x equation:

(-4x+18x+3)/(6x-9)=x

We move all terms to the left:

(-4x+18x+3)/(6x-9)-(x)=0


Domain of the equation: (6x-9)!=0

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

6x!=9

x!=9/6

x!=1+1/2

x∈R


We add all the numbers together, and all the variables

(14x+3)/(6x-9)-x=0

We add all the numbers together, and all the variables

-1x+(14x+3)/(6x-9)=0

We multiply all the terms by the denominator


-1x*(6x-9)+(14x+3)=0

We multiply parentheses

-6x^2+9x+(14x+3)=0

We get rid of parentheses

-6x^2+9x+14x+3=0

We add all the numbers together, and all the variables

-6x^2+23x+3=0

a = -6; b = 23; c = +3;
Δ = b2-4ac
Δ = 232-4·(-6)·3
Δ = 601
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(23)-\sqrt{601}}{2*-6}=\frac{-23-\sqrt{601}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(23)+\sqrt{601}}{2*-6}=\frac{-23+\sqrt{601}}{-12}
$