Expanding the product of two binomials with the area model and the FOIL order, plus squaring a binomial and the difference of two squares — the Grade 9 algebra skill that factorising later reverses. With a diagram, worked examples and a quiz.
In Grade 8 you multiplied a single term into a bracket. Grade 9 raises it to two brackets multiplied together — and this becomes the single most-used skill of the year, because factorising (the next topic) is simply this process run backwards.
1The rule: every term meets every term
Each term in the first bracket must multiply each term in the second. With two terms in each bracket that gives four products, which you then simplify.
- 1First terms: x × x = x2.The F in FOIL.
- 2Outer terms: x × 3 = 3x.The O.
- 3Inner terms: 2 × x = 2x.The I.
- 4Last terms: 2 × 3 = 6.The L.
- 5Add and collect: x2 + 3x + 2x + 6 = x2 + 5x + 6.3x and 2x are like terms.
2Watch the negative signs
- 1x × x = x2.
- 2x × 5 = 5x and (−4) × x = −4x.Keep the minus attached to the 4.
- 3(−4) × 5 = −20.Negative times positive is negative.
- 4x2 + 5x − 4x − 20 = x2 + x − 20.5x − 4x = 1x, written simply as x.
The classic error: writing (x − 4)(x + 5) = x² − 20. Only the first and last terms were multiplied — the two middle products were forgotten. There must always be four products before simplifying.
3Squaring a binomial
- 1(x + 5)2 means (x + 5)(x + 5).Squaring means two identical brackets.
- 2x2 + 5x + 5x + 25.Four products as usual.
- 3= x2 + 10x + 25.The middle term is always 2 × x × 5.
(x + 5)2 is NOT x2 + 25. You cannot square each term separately — the middle term 10x is missing. Test it with x = 1: (1+5)² = 36, but 1 + 25 = 26.
4Difference of two squares
- 1x2 − 7x + 7x − 49.Do the four products as normal.
- 2The middle terms cancel: −7x + 7x = 0.This always happens when the brackets differ only in sign.
- 3= x2 − 49.A difference of two squares — no middle term.
Spotting (a + b)(a − b) saves time: write down a² − b² immediately. You will need this pattern constantly when you start factorising.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Expand: (x + 1)(x + 4)
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x2 + 4x + x + 4 = x2 + 5x + 4 - 2Expand: (x + 3)(x + 6)
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x2 + 9x + 18 - 3Expand: (x − 2)(x + 7)
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x2 + 7x − 2x − 14 = x2 + 5x − 14 - 4Expand: (x − 3)(x − 5)
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x2 − 5x − 3x + 15 = x2 − 8x + 15 - 5Expand: (2x + 1)(x + 4)
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2x2 + 8x + x + 4 = 2x2 + 9x + 4 - 6Expand: (x + 4)2
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x2 + 8x + 16 - 7Expand: (x − 6)2
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x2 − 12x + 36 - 8Expand: (x + 9)(x − 9)
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x2 − 81 (difference of two squares) - 9Expand: (3x + 2)(3x − 2)
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9x2 − 4
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 9 past papers and worksheets on this topic.