Question

How do you multiply `1 / (x(x - 2))+ x /(x - 2) = 10/ x`?

Answer 1

`1 / (x(x - 2))+ x /(x - 2) = 10/ x`

L.C.M of `x(x-2) , (x-2) and x = x(x-2)`

`1 / (x(x - 2))+ (x xx x) /((x - 2) xx x) = (10 xx (x-2))/ (x xx (x-2))`

`(1 + x^2) /( x(x - 2)) = (10x - 20)/ (x (x-2))`

`1 + x^2 = 10x - 20`

`x^2 - 10x +21 = 0`

We can Split the Middle Term of this expression to factorise it
In this technique, if we have to factorise an expression like `ax^2 + bx + c`, we need to think of 2 numbers such that:

`N_1*N_2 = a*c = 1xx21 = 21`

And

`N_1 +N_2 = b = -10`
After trying out a few numbers we get :

`N_1 = -3` and `N_2 =-7`
`-3 xx-7 = 21`, and `(-3) + (-7)= -10`

`x^2 - 10x +21 =x^2 - 3x - 7x+21`

`= x(x-3) - 7(x-3)`
`= (x-3) (x-7)`

the solutions are `x = 3 , x = 7`