Question

How to factorise `4x^2-4x+1`?

Don't see how I can factorise to `(2x-1)^2`? Thank you so much!

Answer 1

`(2x-1)^2`

Explanation:

`"for a quadratic in "color(blue)"standard form"`

`•color(white)(x)ax^2+bx+c ;a!=0`

`"consider the factors of the product ac which sum to b"`

`4x^2-4x+1" is in standard form"`

`"with "a=4,b=-4" and "c=1`

`rArrac=4xx1=4" and - 2 , - 2 sum to - 4"`

`"split the middle term using these factors"`

`4x^2-2x-2x+1larrcolor(blue)"factorise in groups"`

`=color(red)(2x)(2x-1)color(red)(-1)(2x-1)`

`"take out the common factor "(2x-1)`

`=(2x-1)(color(red)(2x-1))`

`rArr4x^2-4x+1=(2x-1)^2`

Answer 2

`4x^2-4x+1 = (2x-1)^2` is a perfect square trinomial...

Explanation:

Given:

`4x^2-4x+1`

This is an example of a perfect square trinomial.

Let's take a look at what happens when you square a binomial using FOIL to help us:

`(A+B)^2 = (A+B)(A+B)`

`color(white)((A+B)^2) = overbrace(A * A)^"First"+overbrace(A * B)^"Outside"+overbrace(B * A)^"Inside"+overbrace(B * B)^"Last"`

`color(white)((A+B)^2) = A^2+AB+AB+B^2`

`color(white)((A+B)^2) = A^2+2AB+B^2`

In our example, note that `4x^2 = (2x)^2` and `1 = 1^2` are both perfect squares. So we might think of putting `A=2x` and `B=1`. That would allow us to find:

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

That gives us `+4x` instead of the `-4x` that we want.

Note however that if we put `B=-1` instead of `B=1` then we still have `B^2 = (-1)(-1) = 1` as we need, but the middle term becomes `AB=(4x)(-1) = -4x` as we also want.

So:

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

More generally, we can write:

`(A-B)^2 = A^2-2AB+B^2`

So given any quadratic in standard form, if the first and last terms are perfect squares and it matches the pattern `A^2+2AB+B^2` or `A^2-2AB+B^2`, then you can recognise it as `(A+B)^2` or `(A-B)^2`.