16/x+8/(x-2)=2

Solution for 16/x+8/(x-2)=2 equation:

16/x+8/(x-2)=2

We move all terms to the left:

16/x+8/(x-2)-(2)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: (x-2)!=0

We move all terms containing x to the left, all other terms to the right

x!=2

x∈R


We calculate fractions


(16x-32)/(x^2-2x)+8x/(x^2-2x)-2=0

We multiply all the terms by the denominator


(16x-32)+8x-2*(x^2-2x)=0

We add all the numbers together, and all the variables

8x+(16x-32)-2*(x^2-2x)=0

We multiply parentheses

-2x^2+8x+(16x-32)+4x=0

We get rid of parentheses

-2x^2+8x+16x+4x-32=0

We add all the numbers together, and all the variables

-2x^2+28x-32=0

a = -2; b = 28; c = -32;
Δ = b2-4ac
Δ = 282-4·(-2)·(-32)
Δ = 528
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:

$\sqrt{\Delta}=\sqrt{528}=\sqrt{16*33}=\sqrt{16}*\sqrt{33}=4\sqrt{33}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(28)-4\sqrt{33}}{2*-2}=\frac{-28-4\sqrt{33}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(28)+4\sqrt{33}}{2*-2}=\frac{-28+4\sqrt{33}}{-4}
$