x-4/x-6=2

Solution for x-4/x-6=2 equation:

x-4/x-6=2

We move all terms to the left:

x-4/x-6-(2)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

x-4/x-8=0

We multiply all the terms by the denominator


x*x-8*x-4=0

We add all the numbers together, and all the variables

-8x+x*x-4=0

Wy multiply elements

x^2-8x-4=0

a = 1; b = -8; c = -4;
Δ = b2-4ac
Δ = -82-4·1·(-4)
Δ = 80
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:

$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{5}}{2*1}=\frac{8-4\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{5}}{2*1}=\frac{8+4\sqrt{5}}{2}
$