Question

How do you simplify `(2)/(x) + (2)/(x-1) - (2)/(x-2)`?

Answer 1

`(2(x^2-4x+2))/(x(x-1)(x-2))`

Explanation:

Before adding/subtracting algebraic fractions we require them to have a `color(blue)"common denominator"`

Multiply numerator/denominator of `2/x" by "(x-1)(x-2)`

Multiply numerator/denominator of `2/(x-1)" by "x(x-2)`

Multiply numerator/denominator of `2/(x-2)" by "x(x-1)`

`• 2/xto(2(x-1)(x-2))/(x(x-1)(x-2))=(2x^2-6x+4)/(x(x-1)(x-2))`

`• 2/(x-1)to(2x(x-2))/(x(x-1)(x-2))=(2x^2-4x)/(x(x-1)(x-2)`

`• 2/(x-2)to(2x(x-1))/(x(x-1)(x-2))=(2x^2-2x)/(x(x-1)(x-2))`

Now the fractions have a common denominator we can add/subtract the numerators leaving the denominator as it is.

Putting the 3 simplifications together.

`rArr(2x^2-6x+4+2x^2-4x-(2x^2-2x))/(x(x-1)(x-2)`

`=(2x^2-8x+4)/(x(x-1)(x-2))`

`=(2(x^2-4x+2))/(x(x-1)(x-2))`