Mathematics • Year 10 • Unit 2 • Lesson 5
Difference of Two Squares — Mixed Challenge
Pull together every idea from Lesson 5: the pattern a² − b² = (a + b)(a − b), coefficients as squares (4x² = (2x)²), HCF-first when terms share factors, and the absolute rule that a² + b² does NOT factor over the reals. Pick the right tool, spot a student's slip, design your own factorisable expression.
1. Mixed problems — choose the right tool
Show your working. 3 marks each
1.1 Factorise x² − 36.
1.2 Factorise 16x² − 1.
1.3 Factorise 49a² − 64b².
1.4 Fully factorise 3x² − 75. (HCF first.)
1.5 Identify which one of the following is a difference of squares and factorise it: (i) x² + 25, (ii) x² − 25, (iii) x² + 5.
1.6 Use the difference-of-squares pattern to quickly compute 73² − 27². (Hint: a + b = 100.)
2. Find the mistake
Another Year 10 student has tried to fully factorise 12x² − 27. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — fully factorise 12x² − 27:
Line 1: HCF(12, 27) = 3
Line 2: 12x² − 27 = 3(4x² − 9)
Line 3: Inside: 4x² − 9 is NOT a difference of squares (4x² is not a perfect square).
Line 4: So the answer is 3(4x² − 9), already fully factorised.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected fully factorised final answer.
Stuck? Is 4x² a perfect square? √(4x²) = 2x — yes, it IS a perfect square.3. Open-ended challenge — design a "double-step" factorising problem
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design an expression of the form k(ax)² − k(by)² (i.e. an expression whose factorising requires two steps: first pull out a common factor, then apply difference of squares).
In your submission, include:
(i) Your starting expression (in expanded form, no brackets — e.g. 8x² − 18y², not 2(4x² − 9y²)).
(ii) Step 1: pull out the HCF.
(iii) Step 2: apply the difference-of-squares pattern inside the bracket.
(iv) The fully factorised form.
(v) A check: expand your fully factorised form back to the starting expression.
Bonus: Use coefficients (a, b) that are NOT 1 — for example, 12x² − 75y² (HCF 3, then (2x + 5y)(2x − 5y)).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — x² − 36
a = x, b = 6. Answer: (x + 6)(x − 6).
1.2 — 16x² − 1
16x² = (4x)², 1 = 1². So a = 4x, b = 1. Answer: (4x + 1)(4x − 1).
1.3 — 49a² − 64b²
49a² = (7a)², 64b² = (8b)². Answer: (7a + 8b)(7a − 8b).
1.4 — 3x² − 75
HCF = 3 → 3(x² − 25). Inside: difference of squares with x and 5.
Fully factorised: 3(x + 5)(x − 5).
1.5 — Which is a difference of squares?
Only (ii) x² − 25 is a difference of squares (two perfect squares with a minus between them).
Factorised: (x + 5)(x − 5).
(i) x² + 25 is a sum of squares — does NOT factorise.
(iii) x² + 5: 5 is not a perfect square (√5 is not rational) and the operation is a sum anyway — does NOT factorise.
1.6 — 73² − 27²
= (73 + 27)(73 − 27) = 100 × 46 = 4600. (Compare: 73² = 5329, 27² = 729; 5329 − 729 = 4600 ✓. The factorising trick is far faster.)
2 — Find the mistake
(a) The mistake is on Line 3 (which leads to the wrong conclusion in Line 4).
(b) 4x² is a perfect square: 4x² = (2x)² because √4 = 2 and √(x²) = x. The student wrongly claimed it isn't, and so missed the second factorising step.
(c) Corrected working:
HCF(12, 27) = 3 → 3(4x² − 9).
Inside: 4x² = (2x)², 9 = 3². Apply difference-of-squares pattern: (2x + 3)(2x − 3).
Fully factorised: 3(2x + 3)(2x − 3).
Lesson § "Spot the Square" explicitly warns that 4x², 9x², 16x², etc. are all perfect squares — the coefficient just needs a whole-number square root.
3 — Open-ended challenge (sample solutions)
Sample 1 — expression 8x² − 18y²
(i) Starting: 8x² − 18y².
(ii) HCF = 2 → 2(4x² − 9y²).
(iii) Inside: 4x² = (2x)², 9y² = (3y)². Pattern → (2x + 3y)(2x − 3y).
(iv) Fully factorised: 2(2x + 3y)(2x − 3y).
(v) Check: 2(2x + 3y)(2x − 3y) = 2(4x² − 9y²) = 8x² − 18y² ✓.
Sample 2 (bonus) — expression 20x² − 45y²
(i) Starting: 20x² − 45y².
(ii) HCF = 5 → 5(4x² − 9y²).
(iii) Inside: pattern with 2x and 3y → (2x + 3y)(2x − 3y).
(iv) Fully factorised: 5(2x + 3y)(2x − 3y).
(v) Check: 5(4x² − 9y²) = 20x² − 45y² ✓.
Marking: 1 for a valid starting expression (HCF > 1 and inside is a difference of squares with non-unit roots), 1 for the HCF step, 1 for the difference-of-squares step, 1 for the verification by expansion. Full marks for any valid construction.