x-3/x-4=2x/x+2

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

x-3/x-4=2x/x+2

We move all terms to the left:

x-3/x-4-(2x/x+2)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: x+2)!=0

x∈R


We get rid of parentheses

x-3/x-2x/x-2-4=0

We multiply all the terms by the denominator


x*x-2x-2*x-4*x-3=0

We add all the numbers together, and all the variables

-8x+x*x-3=0

Wy multiply elements

x^2-8x-3=0

a = 1; b = -8; c = -3;
Δ = b2-4ac
Δ = -82-4·1·(-3)
Δ = 76
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:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{19}}{2*1}=\frac{8-2\sqrt{19}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{19}}{2*1}=\frac{8+2\sqrt{19}}{2}
$