Mathematics • Year 10 • Unit 2 • Lesson 5
Difference of Two Squares, Skill Drill
Build fluency with the pattern a² − b² = (a + b)(a − b). Spot the squares, take the square roots, factor as a sum × difference. Sums of squares a² + b² do NOT factor. One worked example, one guided trace, eight independent problems.
1. I do, fully worked example
Identify the squares, take roots, write as (a + b)(a − b). Read every step.
Problem. Factorise 4x² − 49.
Step 1, Confirm both terms are perfect squares.
4x² = (2x)² ✓ 49 = 7² ✓ Operation between: subtraction ✓
Reason: difference of squares needs TWO squares with a MINUS sign between them.
Step 2, Identify a and b (the square roots).
a = 2x, b = 7
Reason: √(4x²) = 2x and √49 = 7.
Step 3, Apply the pattern a² − b² = (a + b)(a − b).
= (2x + 7)(2x − 7)
Reason: sum factor × difference factor, order of (+) and (−) doesn't matter.
Step 4, Check by expanding.
(2x + 7)(2x − 7) = 4x² − 14x + 14x − 49 = 4x² − 49 ✓
Answer: 4x² − 49 = (2x + 7)(2x − 7).
2. We do, fill in the missing steps
This time HCF first, then difference of squares. Fill in each blank. 5 marks
Problem. Fully factorise 3x² − 27.
Step 1, Is this directly a difference of squares? Check whether 3x² is a perfect square. (√3 is not a whole number.) Answer: ____ (yes / no).
Step 2, Look for a common factor first. HCF(3x², 27) = ____.
Step 3, Pull out the HCF:
3x² − 27 = ____ ( ____ − ____ )
Step 4, Now look inside the bracket, is THIS a difference of squares?
Inside: x² − 9. a = ____, b = ____ .
Apply pattern: ( ____ + ____ )( ____ − ____ )
Step 5, Write the fully factorised form (don't forget the 3 outside):
3x² − 27 = ________________________
3. You do, independent practice
Show your working. First four are foundation. Next two are standard. Last two are extension.
Foundation, basic difference of squares
3.1 Factorise x² − 16. 1 mark
3.2 Factorise x² − 81. 1 mark
3.3 Factorise x² − 100. 1 mark
3.4 Factorise 25 − y². 1 mark
Standard, coefficients as squares, two letters
3.5 Factorise 9a² − 16b². 2 marks
3.6 Fully factorise 2x² − 50. (HCF first!) 2 marks
Extension, push your thinking
3.7 Fully factorise 5x² − 45y². 3 marks
3.8 A student writes x² + 25 = (x + 5)(x + 5). In one sentence, explain what is wrong, then state correctly whether x² + 25 can be factorised at all (using real numbers). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (3x² − 27)
Step 1: no3x² is not a perfect square.
Step 2: HCF(3x², 27) = 3.
Step 3: 3x² − 27 = 3(x² − 9).
Step 4: x² − 9 IS a difference of squares. a = x, b = 3. Pattern: (x + 3)(x − 3).
Step 5: 3x² − 27 = 3(x + 3)(x − 3).
3.1, x² − 16
a = x, b = 4. Answer: (x + 4)(x − 4).
3.2, x² − 81
a = x, b = 9. Answer: (x + 9)(x − 9).
3.3, x² − 100
a = x, b = 10. Answer: (x + 10)(x − 10).
3.4-25 − y²
a = 5, b = y. Answer: (5 + y)(5 − y).
The order matters here, write the larger-rooted term first inside both factors.
3.5-9a² − 16b²
9a² = (3a)²; 16b² = (4b)². So a = 3a (sorry, call it the "first root"), b-root = 4b.
Answer: (3a + 4b)(3a − 4b).
3.6-2x² − 50 (HCF first!)
HCF = 2 → 2(x² − 25). Now x² − 25 is a difference of squares: (x + 5)(x − 5).
Fully factorised: 2(x + 5)(x − 5).
3.7-5x² − 45y² (fully)
HCF = 5 → 5(x² − 9y²). Inside: a = x, b = 3y. So x² − 9y² = (x + 3y)(x − 3y).
Fully factorised: 5(x + 3y)(x − 3y).
3.8, What's wrong with x² + 25 = (x + 5)(x + 5)?
The student's right-hand side (x + 5)(x + 5) expands to x² + 10x + 25, not x² + 25, so the equation is false. Moreover, x² + 25 is a sum of squares, and sums of squares do not factorise over the real numbers. So the correct answer is: x² + 25 is already in simplest form and cannot be factorised using real numbers.