Mathematics • Year 10 • Unit 2 • Lesson 5

Difference of Two Squares in the Real World

Apply a² − b² = (a + b)(a − b) to area-difference problems, mental-arithmetic shortcuts, ring-shaped designs and a fast way to multiply numbers near a round figure. Then explain why a sum of squares is the most common trap.

Apply · Real-World Maths

1. Word problems

Each problem uses the difference-of-squares pattern from Lesson 5. Show your working.

1.1 — Picture frame area. A square painting of side x cm hangs inside a square frame of side (x + 4) cm.

(a) Write an expression for the area of just the frame border (frame area − painting area).
(b) Expand and simplify to a form with no brackets.
(c) Re-factorise using a common factor and the difference-of-squares pattern where possible. 4 marks

Stuck? Border area = (x + 4)² − x². Expand the square first, then collect — the x² cancels.

1.2 — Mental arithmetic shortcut. The difference-of-squares pattern lets you compute things like 51 × 49 in your head: 51 × 49 = (50 + 1)(50 − 1) = 50² − 1² = 2500 − 1 = 2499.

Use this trick to evaluate:
(a) 102 × 98
(b) 21 × 19
(c) 99² − 98² (use the pattern in reverse: a² − b² = (a + b)(a − b)). 3 marks

Stuck on (c)? 99² − 98² = (99 + 98)(99 − 1) — no wait — (99 + 98)(99 − 98) = 197 × 1 = 197.

1.3 — Difference of two square gardens. A council removes a small square lawn of side 3 m from the corner of a larger square lawn of side x m to make a flower bed.

(a) Write an expression for the remaining lawn area.
(b) Factorise the expression using the difference-of-squares pattern.
(c) If x = 8, calculate the remaining lawn area in m². 3 marks

Stuck? Remaining = x² − 9 = (x + 3)(x − 3). Substitute x = 8 at the end.

1.4 — Two square paddocks. A farmer compares a square paddock of side 2a m with a square paddock of side b m. The "difference in area" question asks 4a² − b².

(a) Confirm whether 4a² − b² is a difference of squares (state a and b in the pattern).
(b) Factorise it.
(c) If a = 6 and b = 5, calculate the difference in area. 3 marks

Stuck? 4a² = (2a)². So the pattern uses (2a) and b. Factorised: (2a + b)(2a − b).

1.5 — A circular-style "ring" pulled out. The expression 50 − 2x² appears in a manufacturer's specification for two square plates of cardboard removed from a 50 cm² template.

(a) Pull out the HCF first.
(b) Use the difference-of-squares pattern on what's left.
(c) State for which values of x (in centimetres, positive whole numbers) the expression equals zero. 4 marks

Stuck? 50 − 2x² = 2(25 − x²) = 2(5 + x)(5 − x). Zero when x = 5.

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A classmate writes: "x² + 16 factorises to (x + 4)(x + 4)." Using everything from Lesson 5, explain (i) why their answer expands to something different (do the FOIL check), (ii) why a sum of squares like x² + 16 cannot be factorised using real numbers, and (iii) what kind of expression CAN be factorised by the difference-of-squares pattern. Use the words "difference" and "sum of squares" somewhere in your answer.

Stuck? Revisit lesson § "Sums Stay Put" — a sum of squares stays unchanged; only a difference of two squares can be split into (a + b)(a − b).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Picture frame border

(a) Border = (x + 4)² − x².
(b) Expand (x + 4)² = x² + 8x + 16. Subtract x²: 8x + 16 cm².
(c) Re-factorise: 8(x + 2) cm². (Direct difference of squares with the (x + 4)² − x² form: ((x + 4) + x)((x + 4) − x) = (2x + 4)(4) = 4(2x + 4) = 8(x + 2). Both routes match.)

1.2 — Mental arithmetic shortcut

(a) 102 × 98 = (100 + 2)(100 − 2) = 100² − 2² = 10000 − 4 = 9996.
(b) 21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399.
(c) 99² − 98² = (99 + 98)(99 − 98) = 197 × 1 = 197.

1.3 — Difference of two square gardens

(a) Remaining = x² − 9 m².
(b) Pattern with a = x, b = 3 → (x + 3)(x − 3).
(c) When x = 8: (8 + 3)(8 − 3) = 11 × 5 = 55 m². (Check: 64 − 9 = 55 ✓.)

1.4 — Two square paddocks

(a) 4a² = (2a)² so yes — difference of squares with "a-root" = 2a and "b-root" = b.
(b) (2a + b)(2a − b).
(c) When a = 6, b = 5: (12 + 5)(12 − 5) = 17 × 7 = 119 m². (Check: 4(36) − 25 = 144 − 25 = 119 ✓.)

1.5 — Manufacturer's expression 50 − 2x²

(a) HCF = 2 → 2(25 − x²).
(b) Inside is a difference of squares with a = 5, b = x: 2(5 + x)(5 − x).
(c) The expression equals zero when one of the factors is zero. 2 ≠ 0; 5 + x = 0 → x = −5 (not a positive cm); 5 − x = 0 → x = 5. So the expression is zero when x = 5 cm.

2.1 — Explain your thinking (sample response)

(i) FOIL check: (x + 4)(x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16 — that's not x² + 16, so the classmate's claim is wrong. (ii) x² + 16 is a sum of squares, not a difference, and a sum of squares cannot be factorised using real numbers — there are no two real numbers (a, b) such that (x + a)(x + b) expands to x² + 16 without a middle term. (iii) Only a difference of two squares — an expression of the form a² − b², with two perfect squares and a minus sign between them — can be split into (a + b)(a − b).

Marking: 1 for the correct FOIL expansion of (x + 4)², 1 for naming "sum of squares" as the obstacle, 1 for the correct shape that DOES factorise (a² − b²), 1 for using both required words.