Question

How do you solve `x^2(a^2+2ab+b^2) = x(a+b)` by factoring?

Answer 1

`x = 0" "` or `" "x = 1/(a+b)`

Explanation:

First, notice that you can write

`(a^2 + 2ab + b^2)`

as the square of `(a + b)`

`a^2 + 2ab + b^2 = (a + b)^2`

Your equation can now be written as

`x^2 * (a+b)^2 = x * (a + b)`

You can simplify this equation by dividing both sides by `(a+b)`

`(x^2 * (a + b)^color(red)(cancel(color(black)(2))))/color(red)(cancel(color(black)((a+b)))) = (x * color(red)(cancel(color(black)((a+b)))))/color(red)(cancel(color(black)((a+b))))`

`x^2 * (a + b) = x`

Move all the terms to one side of the equation to get

`(a+b) * x^2 - x = 0`

`x * [(a+b)x - 1] = 0`

This equation will have two solutions, `x = color(green)(0)` and

`(a+b)x - 1= 0 implies x = color(green)(1/(a+b))`