KS3 · Maths

Difference of two squares

Multiply out two brackets and you usually get three or four terms. But some pairs are special: the middle just vanishes. Soon you'll spot them before multiplying anything.

Algebra · Multiplying two brackets

Two terms, three terms or four?

Don't multiply yet. Just look at each pair of brackets and predict: how many terms will the simplified answer have?

Still to sort

Two terms (0)

Two of the partial products cancel to nothing.

Where the line is: Needs the same term in both brackets AND a zero pair, like +5 and −5. Then the middle products are a zero pair too, and vanish.

Three terms (0)

Two partial products are like terms and combine into one.

Where the line is: The like terms have the same sign, like −5x and −5x, so they add together instead of cancelling.

Four terms (0)

None of the partial products are like terms, so nothing combines.

Where the line is: Different letters in the two brackets, like x and y, or a and b, mean the partial products can't be collected.

7 of 7 still to sort.

Every pair of brackets multiplies out to four partial products — each term in one bracket times each term in the other. The question is how many terms are still standing once you collect like terms.

Watch out: Opposite signs on their own are not enough, and neither is a repeated bracket. You need BOTH: one term the same in each bracket, AND the other terms a zero pair.

Why the middle vanishes

Build (a + b)(a − b) yourself

Here's an area model: one bracket down the side, the other along the top. Each cell is its row times its column. Fill all four cells, then collect them into one answer.

Grid: each cell is its row times its column
a−b
a
+b

Type x² as x^2 if you cannot type ². Spaces do not matter.

Predict, then check

Same shape as before — but now there's a number stuck to the letter.

What is (3a + b)(3a − b)?

Spot the slip

Where does this go wrong?

Expand (x + 4)² and say how many terms the answer has.

A student's answer — which line goes wrong?

Now run it backwards

Find the brackets

Write 25a² − 9b² as the product of two brackets.

  1. 25a² − 9b² is one square take away another, so it is a difference of two squares.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Four partial products go in. Sometimes only two terms come out.

What you need to know

  • A binomial is an expression with exactly two unlike terms, such as x + 5 or 3a − b.
  • Multiplying two binomials always gives four partial products: each term in one bracket times each term in the other.
  • A zero pair is two terms that add to zero, like +6x and −6x.
  • (a + b)(a − b) = a² − b², where a and b can be any terms.

The big picture

Multiplying two binomials always makes four partial products. How many terms survive depends on like terms: no like terms leaves four; two like terms that combine leaves three; and when one term is the same in both brackets and the other terms are a zero pair, the middle products cancel and leave a² − b², the difference of two squares. The structure runs backwards too: 25a² − 9b² = (5a + 3b)(5a − 3b).

Key points

1No like terms among the partial products means four terms: (x + 3)(y + 3) = xy + 3x + 3y + 9.
2Two like terms that combine means three terms: (x + 4)(x + 6) = x² + 10x + 24.
3One term the same in both brackets plus a zero pair means the middle cancels, leaving two terms: (x + 5)(x − 5) = x² − 25.
4Opposite signs alone are not enough: (a + 4)(b − 4) = ab − 4a + 4b − 16 has nothing to cancel.
5Square the whole term: (3a + b)(3a − b) = 9a² − b² = (3a)² − b².
6A squared bracket is not a difference of two squares: (x + 4)² = x² + 8x + 16.
7To work backwards, square-root each term: 25a² − 9b² = (5a + 3b)(5a − 3b).

Worked example

Problem

Expand and simplify (6a + 5b)(6a − 5b).

⚠ Watch out

Thinking any opposite signs give a difference of two squares. In (a + 4)(b − 4), the +4 and −4 multiply different letters, so −4a and +4b can't cancel: you get ab − 4a + 4b − 16. You need the same term in both brackets too.

🧠

Memory hook

Same, opposite, gone: the SAME term in both brackets, OPPOSITE signs on the other term, and the middle is GONE.

✓

Check yourself

Before expanding, predict how many terms (k + 9)(k − 9), (k + 9)(k + 9) and (k + 9)(m − 9) give. (Two, three, four: k² − 81 is the two.)

Flashcards

(14)
What is a binomial?
An algebraic expression with exactly two unlike terms, such as x + 5 or 3a − b.
What is a partial product?
One of the multiplication results that lead up to the whole product — for example, each cell of an area model.
How many partial products do you get when you multiply two binomials?
Always four: each term in one bracket multiplies each term in the other.
What are like terms?
Terms with exactly the same letters to the same powers, like 6x and 4x. They can be collected into one term.
What is a zero pair?
Two terms that are additive inverses, like +6x and −6x. They add to zero.
Why do most products of two binomials give three terms?
Two of the four partial products are like terms, so they combine into one term.
What is the difference of two squares?
An expression of the form a² − b²: one square subtracted from another.
(a + b)(a − b) = ?
a² − b². The middle products +ab and −ab are a zero pair and cancel.
What two things must the brackets have to give a difference of two squares?
One term the same in each bracket, and the other terms a zero pair (one positive, one negative).
Why isn't (a + 4)(b − 4) a difference of two squares?
The +4 and −4 multiply different letters, so −4a and +4b are not like terms and nothing cancels.
Does it matter which bracket comes first, (x − 8)(x + 8) or (x + 8)(x − 8)?
No — both give x² − 64.
What does (a + b)² expand to?
a² + 2ab + b² — three terms, because the two ab products add instead of cancelling.
Which two brackets multiply to give x² − 16?
(x + 4)(x − 4). The −16 comes from −4 multiplied by +4.
What happens if BOTH pairs of terms in the brackets are zero pairs?
Nothing is the same in both brackets, so the middle products don't cancel — the product has three terms, not two.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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