(3x+6)/(x^2-36)=6

Solution for (3x+6)/(x^2-36)=6 equation:

(3x+6)/(x^2-36)=6

We move all terms to the left:

(3x+6)/(x^2-36)-(6)=0


Domain of the equation: (x^2-36)!=0

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

x^2!=36

x^2!=36/

x^2!=√1/0

x!=1

x∈R


We multiply all the terms by the denominator


(3x+6)-6*(x^2-36)=0

We multiply parentheses

-6x^2+(3x+6)+216=0

We get rid of parentheses

-6x^2+3x+6+216=0

We add all the numbers together, and all the variables

-6x^2+3x+222=0

a = -6; b = 3; c = +222;
Δ = b2-4ac
Δ = 32-4·(-6)·222
Δ = 5337
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:

$\sqrt{\Delta}=\sqrt{5337}=\sqrt{9*593}=\sqrt{9}*\sqrt{593}=3\sqrt{593}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-3\sqrt{593}}{2*-6}=\frac{-3-3\sqrt{593}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+3\sqrt{593}}{2*-6}=\frac{-3+3\sqrt{593}}{-12}
$