Lesson 14: Factorising using Difference of Two Squares

Introduction

We're going to be talking about expressions like \(~4x^2-25~\) and how to
factorise them. You might think this looks like an expression from
Lesson 10: Factorising to a single bracket. But the highest common factor of 4
and 25 is a pointless 1! Plus there's only an \(~x~\) part in one of the terms.
Well, maybe we can use the methods we learned in Lessons 12 and 13: Factorising to
double brackets. Nope! Where's the \(~x~\) term, what's the value of \(~b~\)?
Actually, this is just a special case which is fortunately quite easy to factorise.
Allow me to explain.

SECTION A

I'm gonna get straight down to the nitty-gritty, if you don't mind.

Basically, if you've got an expression \(~■-□~\) and you notice that \(~■~\)
and \(~□~\) are squares (I love the fact that they are actually squares!),
you can just rewrite the expression like this:

\((\sqrt{■}+\sqrt{□})(\sqrt{■}-\sqrt{□})\)

This rule is usually written as \(~(a^2-b^2)=(a+b)(a-b)~\) but I prefer my way. Each to his (or her) own!

Example 1

Factorise \(~4x^2-25~\)

I've noticed that \(~4x^2~\) and \(~25~\) are both squares!

\(\sqrt{4x^2}=2x~\) and \(~\sqrt{25}=5\)

Answer: \(~(2x+5)(2x-5)\)

About the question

Both terms have to be squares because you have to be able to square root them!

One term has to be subtracted from the other. In other words, the expression has to be a difference ...
a difference of two squares! Hey, that's where the name comes from!

About the answer

The answer is always two brackets, one with a \(~+~\) and one with a \(~-~\) (it doesn't matter which sign goes in which bracket).

To use this method, you have to be able to square root algebraic terms.

How to square root terms

To square root a term, just square root each part separately.

Start with the number (e.g. \(~\sqrt{64}=8~\)).
If the number is not a square number, you cannot use the difference of two squares.

To square root a letter part, just half the power (e.g. \(~\sqrt{x^6}=x^3~\)).
If the power of any letter is an odd number, it means the term is not a square so you can't use difference of two squares.

Example 2

Factorise \(~9x^4y^2-64x^2~\)

I notice this is the difference of two squares!

\(\sqrt{9x^4y^2}=3x^2y~\) and \(~\sqrt{64x^2}=8x\)

Answer: \(~(3x^2y+8x)(3x^2y-8x)\)

Example 3

Factorise \(~x^4+49~\)

No can do! It's called the difference
of two squares, NOT the sum of two squares!

Example 4

Factorise \(~5x^2-49x^4y^3~\)

Not a chance! 5 is not a square number and \(~y^3~\) is not a square, either.
I know this because the power is an odd number. Ooh, wait a minute!
I can't use the difference of two squares method but ...

... I can factorise to a single bracket just like I did in Lesson 10! So ...

\(~5x^2-49x^4y^3~=~x^2(5-49x^2y^3)\)

In the following practice questions, most of the expressions can be factorised
using the difference of two squares method, some can't but they can be factorised
to a single bracket, some can't be factorised at all. This will give you valuable
experience of not only using difference of two squares but also recognising when
to use it.

Over to you!

Practise to master

SECTION A

Factorise the following expressions however you can (if you can).

01) \(~x^2-4~\)

I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{4}=2\)
Answer: \((x+2)(x-2)\)

02) \(~x^2+9~\)

NOT difference of two squares due to \(~+~\) sign!
Can't factorise to a single bracket either.
Answer: Will NOT factorise!

03) \(~x^2-100~\)

I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{100}=10\)
Answer: \((x+10)(x-10)\)

04) \(~x^2-25~\)

I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{25}=5\)
Answer: \((x+5)(x-5)\)

05) \(~x^2-64~\)

I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{64}=8\)
Answer: \((x+8)(x-8)\)

06) \(~x^2-35~\)

NOT difference of two squares since 35 is not a square!
Answer: Will NOT factorise!

07) \(~4x^2-1~\)

I notice this is the difference of two squares!
\(\sqrt{4x^2}=2x~\) and \(~\sqrt{1}=1\)
Answer: \((2x+1)(2x-1)\)

08) \(~25x^2-16~\)

I notice this is the difference of two squares!
\(\sqrt{25x^2}=5x~\) and \(~\sqrt{16}=4\)
Answer: \((5x+4)(5x-4)\)

09) \(~36x^2-2~\)

NOT difference of two squares since 2 is not a square!
This will factorise to a single bracket, though.
Answer: \(2(18x^2-1)\)

10) \(~64x^2-81~\)

I notice this is the difference of two squares!
\(\sqrt{64x^2}=8x~\) and \(~\sqrt{81}=9\)
Answer: \((8x+9)(8x-9)\)

11) \(~9x^2+4~\)

NOT difference of two squares due to \(~+~\) sign!
Answer: Will NOT factorise!

12) \(~16x^2-9~\)

I notice this is the difference of two squares!
\(\sqrt{16x^2}=4x~\) and \(~\sqrt{9}=3\)
Answer: \((4x+3)(4x-3)\)

13) \(~12x^2-4~\)

NOT difference of two squares since 12 is not a square!
This will factorise to a single bracket, though.
Answer: \(4(3x^2-1)\)

13) \(~100x^2-9x~\)

NOT difference of two squares since \(~x~\) is not a square!
This will factorise to a single bracket, though.
Answer: \(x(100x-9)\)

15) \(~25x^2-121~\)

I notice this is the difference of two squares!
\(\sqrt{25x^2}=5x~\) and \(~\sqrt{121}=11\)
Answer: \((5x+11)(5x-11)\)

16) \(~81x^2-100~\)

I notice this is the difference of two squares!
\(\sqrt{81x^2}=9x~\) and \(~\sqrt{100}=10\)
Answer: \((9x+10)(9x-10)\)

17) \(~64x^2-y^2~\)

I notice this is the difference of two squares!
\(\sqrt{64x^2}=8x~\) and \(~\sqrt{y^2}=y\)
Answer: \((8x+y)(8x-y)\)

18) \(~4x^2-81y^2~\)

I notice this is the difference of two squares!
\(\sqrt{4x^2}=2x~\) and \(~\sqrt{81y^2}=9y\)
Answer: \((2x+9y)(2x-9y)\)

19) \(~9x^2y^2-49z^2~\)

I notice this is the difference of two squares!
\(\sqrt{9x^2y^2}=3xy~\) and \(~\sqrt{49z^2}=7z\)
Answer: \((3xy+7z)(3xy-7z)\)

20) \(~25x^2y^4-36z^2~\)

I notice this is the difference of two squares!
\(\sqrt{25x^2y^4}=5xy^2~\) and \(~\sqrt{36z^2}=6z\)
Answer: \((5xy^2+6z)(5xy^2-6z)\)

SECTION B

We're nearly there, folks! I just want to throw something else your way -
it's for your own good! It may be that a combination of two methods is required.

Example 1

Factorise \(~4x^2-16~\)

I notice this is the difference of two squares!

\(\sqrt{4x^2}=2x~\) and \(~\sqrt{16}=4\)

Answer: \(~(2x+4)(2x-4)\)

I now notice that a factor of 2 can be taken out of each bracket

Final answer: \(~2(x+2)2(x-2)~=~4(x+2)(x-2)\)

I think it's slightly easier to do this the other way round, though ...

Example 1 (again)

Factorise \(~4x^2-16~\)

I can factorise this to a single bracket

\(~4(x^2-4)~\)

I notice this is the difference of two squares!

\(\sqrt{x^2}=x~\) and \(~\sqrt{4}=2\)

Answer: \(4(x+2)(x-2)\)

Yeah, that's better!

Example 2

Factorise \(~8x^2-18~\)

I can factorise this to a single bracket

\(~2(4x^2-9)~\)

I notice this is the difference of two squares!

\(\sqrt{4x^2}=2x~\) and \(~\sqrt{9}=3\)

Answer: \(2(2x+3)(2x-3)\)

These shouldn't be too much of a problem for you.

Practise to master

SECTION B

Factorise the following expressions.

01) \(~9x^2-81~\)

Factorise to a single bracket
\(9(x^2-9)\)
I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{9}=3\)
Answer: \(9(x+3)(x-3)\)

02) \(~36x^2-144~\)

Factorise to a single bracket
\(36(x^2-4)\)
I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{4}=2\)
Answer: \(36(x+2)(x-2)\)

03) \(~64x^2-100~\)

Factorise to a single bracket
\(4(16x^2-25)\)
I notice this is the difference of two squares!
\(\sqrt{16x^2}=4x~\) and \(~\sqrt{25}=5\)
Answer: \(4(4x+5)(4x-5)\)

04) \(~4x^2-36~\)

Factorise to a single bracket
\(4(x^2-9)\)
I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{9}=3\)
Answer: \(4(x+3)(x-3)\)

05) \(~81x^2-144~\)

Factorise to a single bracket
\(9(9x^2-16)\)
I notice this is the difference of two squares!
\(\sqrt{9x^2}=3x~\) and \(~\sqrt{16}=4\)
Answer: \(9(3x+4)(3x-4)\)

06) \(~8x^2-50~\)

Factorise to a single bracket
\(2(4x^2-25)\)
I notice this is the difference of two squares!
\(\sqrt{4x^2}=2x~\) and \(~\sqrt{25}=5\)
Answer: \(2(2x+5)(2x-5)\)

07) \(~27x^2-48~\)

Factorise to a single bracket
\(3(9x^2-16)\)
I notice this is the difference of two squares!
\(\sqrt{9x^2}=3x~\) and \(~\sqrt{16}=4\)
Answer: \(3(3x+4)(3x-4)\)

08) \(~16x^2-64~\)

Factorise to a single bracket
\(16(x^2-4)\)
I notice this is the difference of two squares!
\(\sqrt{x^2}=x~\) and \(~\sqrt{4}=2\)
Answer: \(16(x+2)(x-2)\)

09) \(~100x^2-16y^2~\)

Factorise to a single bracket
\(4(25x^2-4y^2)\)
I notice this is the difference of two squares!
\(\sqrt{25x^2}=5x~\) and \(~\sqrt{4y^2}=2y\)
Answer: \(4(5x+2y)(5x-2y)\)

10) \(~72x^2-50y^2~\)

Factorise to a single bracket
\(2(36x^2-25y^2)\)
I notice this is the difference of two squares!
\(\sqrt{36x^2}=6x~\) and \(~\sqrt{25y^2}=5y\)
Answer: \(2(6x+5y)(6x-5y)\)

11) \(~16a^2b^2-144c^2~\)

Factorise to a single bracket
\(16(a^2b^2-9c^2)\)
I notice this is the difference of two squares!
\(\sqrt{a^2b^2}=ab~\) and \(~\sqrt{9c^2}=3c\)
Answer: \(16(ab+3c)(ab-3c)\)

12) \(~144a^2b^2-9b^2~\)

Factorise to a single bracket
\(9b^2(16a^2-1)\)
I notice this is the difference of two squares!
\(\sqrt{16a^2}=4a~\) and \(~\sqrt{1}=1\)
Answer: \(9b^2(4a+1)(4a-1)\)