Question

How do you solve `(2(x-2))/(x^2-10x+16)=2/(x+2)`?

Answer 1

`x in {cancel"O"}`

(no solutions)

Explanation:

First, let's factor `x^2-10x+16`.

We need two factors of 16 which add up to -10.

`-2 and -8` work, so our factored polynomial is:

`(x-2)(x-8)`

Now we can solve the equation:

`(2(x-2))/(x^2-10x+16)=2/(x+2)`

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

`(color(blue)cancel(color(black)2)color(red)cancel(color(black)((x-2))))/(color(red)cancel(color(black)((x-2)))(x-8)) = color(blue)cancel(color(black)2)/(x+2)`

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

Now take the reciprocal of both sides:

`x-8 = x+2`

`-8 = 2`

This is impossible, so there are no solutions.

Answer 2

No solution

Explanation:

`(2(x-2))/(x^2-10x+16)=2/(x+2)`

`:.(2(cancel(x-2)^1))/((cancel(x-2)^1)(x-8))=2/(x+2)`

`:.2/(x-8)=2/(x+2)`

multiply both sides by (x-8)(x+2)

`:.2(x+2)=2(x-8)`

`:.2x+4=2x-16`

`:.2x-2x=-16-4`

`:.0=-20`

No solution