How to factorise 4x^2-4x+14x24x+1?

Don't see how I can factorise to (2x-1)^2(2x1)2? Thank you so much!

2 Answers
Apr 14, 2018

(2x-1)^2(2x1)2

Explanation:

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

•color(white)(x)ax^2+bx+c ;a!=0xax2+bx+c;a0

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

4x^2-4x+1" is in standard form"4x24x+1 is in standard form

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

rArrac=4xx1=4" and - 2 , - 2 sum to - 4"ac=4×1=4 and - 2 , - 2 sum to - 4

"split the middle term using these factors"split the middle term using these factors

4x^2-2x-2x+1larrcolor(blue)"factorise in groups"4x22x2x+1factorise in groups

=color(red)(2x)(2x-1)color(red)(-1)(2x-1)=2x(2x1)1(2x1)

"take out the common factor "(2x-1)take out the common factor (2x1)

=(2x-1)(color(red)(2x-1))=(2x1)(2x1)

rArr4x^2-4x+1=(2x-1)^24x24x+1=(2x1)2

Apr 14, 2018

4x^2-4x+1 = (2x-1)^24x24x+1=(2x1)2 is a perfect square trinomial...

Explanation:

Given:

4x^2-4x+14x24x+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)(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.