Question

How do you factor `x^2-2x+1`?

Answer 1

This is a perfect square trinomial:

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

Explanation:

In general, you can recognise these perfect square trinomials, because they take the form `a^2+2ab+b^2 = (a+b)^2` or `a^2-2ab+b^2 = (a-b)^2`

In this particular case, there's a little trick to help spot it:

The coefficients of your quadratic - ignoring the signs - are `1`, `2` and `1`. Does the sequence `1 2 1` ring any bells? Well `121 = 11*11 = 11^2` and `x^2-2x+1 = (x-1)^2`, where the coefficients of `x-1` - ignoring signs - are `1` and `1`.

This works with a few other trinomials, like:

`x^2+6x+9 = (x+3)^2` like `169 = 13^2`

It only works if the numbers are small enough, but it can help.