3x+6/x=9x+12

Solution for 3x+6/x=9x+12 equation:

3x+6/x=9x+12

We move all terms to the left:

3x+6/x-(9x+12)=0


Domain of the equation: x!=0

x∈R


We get rid of parentheses

3x+6/x-9x-12=0

We multiply all the terms by the denominator


3x*x-9x*x-12*x+6=0

We add all the numbers together, and all the variables

-12x+3x*x-9x*x+6=0

Wy multiply elements

3x^2-9x^2-12x+6=0

We add all the numbers together, and all the variables

-6x^2-12x+6=0

a = -6; b = -12; c = +6;
Δ = b2-4ac
Δ = -122-4·(-6)·6
Δ = 288
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:

$\sqrt{\Delta}=\sqrt{288}=\sqrt{144*2}=\sqrt{144}*\sqrt{2}=12\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-12\sqrt{2}}{2*-6}=\frac{12-12\sqrt{2}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+12\sqrt{2}}{2*-6}=\frac{12+12\sqrt{2}}{-12}
$