X+4=6x-12/x-2=3x+12

Solution for X+4=6x-12/x-2=3x+12 equation:

X+4=6X-12/X-2=3X+12

We move all terms to the left:

X+4-(6X-12/X-2)=0


Domain of the equation: X-2)!=0

X∈R


We get rid of parentheses

X-6X+12/X+2+4=0

We multiply all the terms by the denominator


X*X-6X*X+2*X+4*X+12=0

We add all the numbers together, and all the variables

6X+X*X-6X*X+12=0

Wy multiply elements

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

We add all the numbers together, and all the variables

-5X^2+6X+12=0

a = -5; b = 6; c = +12;
Δ = b2-4ac
Δ = 62-4·(-5)·12
Δ = 276
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:

$\sqrt{\Delta}=\sqrt{276}=\sqrt{4*69}=\sqrt{4}*\sqrt{69}=2\sqrt{69}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{69}}{2*-5}=\frac{-6-2\sqrt{69}}{-10}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{69}}{2*-5}=\frac{-6+2\sqrt{69}}{-10}
$