x-3/x-2=9/x+6

Solution for x-3/x-2=9/x+6 equation:

x-3/x-2=9/x+6

We move all terms to the left:

x-3/x-2-(9/x+6)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: x+6)!=0

x∈R


We get rid of parentheses

x-3/x-9/x-6-2=0

We multiply all the terms by the denominator


x*x-6*x-2*x-3-9=0

We add all the numbers together, and all the variables

-8x+x*x-12=0

Wy multiply elements

x^2-8x-12=0

a = 1; b = -8; c = -12;
Δ = b2-4ac
Δ = -82-4·1·(-12)
Δ = 112
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:

$\sqrt{\Delta}=\sqrt{112}=\sqrt{16*7}=\sqrt{16}*\sqrt{7}=4\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{7}}{2*1}=\frac{8-4\sqrt{7}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{7}}{2*1}=\frac{8+4\sqrt{7}}{2}
$