Algebraic expressions · Ages 13–14

Factorising a difference of two squares worksheet

Part of: Factorisation

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Algebraic expressions · Ages 13–14

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Write each answer in the space. For word problems, show your working and write a sentence answer.

8.x² − 49 can be factorised as (x + 7)(x − 7).
True or false?
Answer:
9.Factorise 4x² − 49. What do you get?
- A. (2x + 7)²
- B. (2x + 7)(2x − 7)
- C. (2x − 7)²
- D. (4x + 49)(4x − 49)
Choose the right answer
Answer:
10.Factorise 4x² − 9y². What do you get?
- A. (2x + 3y)(2x − 3y)
- B. (3x + 2y)(3x − 2y)
- C. (2x + 3y)²
- D. (2x − 3y)²
Choose the right answer
Answer:
11.(x + 10)² − 25 can be factorised as (x + 15)(x + 10).
True or false?
Answer:
12.Factorise 16x² − 25y². What do you get?
- A. (5x + 4y)(5x − 4y)
- B. (4x + 5y)(4x − 5y)
- C. (4x − 5y)²
- D. (4x + 5y)²
Choose the right answer
Answer:

13.Factorise 5x² − 20y² completely: 5x² − 20y² = (x + y)(x − y).
Fill in each blank
Working:
Answer:
14.Two square rugs have sides 57 cm and 53 cm long. Using the difference of two squares, their areas differ by cm².
Working:
Answer:
15.Factorise 5x³ − 180x completely. Put the steps in order.
- A. 5x³ − 180x = 5x(x² − 36)
- B. = 5x(x + 6)(x − 6)
- C. = 5x(x² − 6²)
Write the letters in the right order
Working:
Answer:
16.Which of these can't be factorised as a difference of two squares?
- A. 64x² − 9
- B. 9 − x²
- C. x² + 9
- D. x²y² − 64
Choose the right answer
Working:
Answer:
17.(x + 7)² − (x + 3)² factorises as (2x + 10)(10).
True or false?
Working:
Answer:
18.A square lawn has sides x m long, and a square pond 2 m on a side is dug in one corner. The area left, (x² − 4) m², can be written (x + )(x − ) m².
Fill in each blank
Working:
Answer:

Show the answers and working

Factorising a difference of two squares worksheet: Answers and working

1. 5, 5
25 = 5², so x² − 25 = x² − 5² = (x + 5)(x − 5).
2. 9, 9
81 = 9², so x² − 81 = x² − 9² = (x + 9)(x − 9).
3. 1800
55² − 35² = (55 + 35)(55 − 35) = 90 × 20 = 1800.
4. 5, 1, 1
Take out the common factor 5 first: 5(x² − 1); then the difference of squares: 5(x + 1)(x − 1).
5. 3120
73² − 47² = (73 + 47)(73 − 47) = 120 × 26 = 3120.
6. 49, 7, 7
x⁴ − 2401 = (x²)² − 49² = (x² + 49)(x² − 49), and x² − 49 factorises again into (x + 7)(x − 7).
7. 4, 2, 2
x⁴ − 16 = (x²)² − 4² = (x² + 4)(x² − 4), and x² − 4 factorises again into (x + 2)(x − 2).

8. True
(x + 7)(x − 7) = x² − 49. Only two squares with a minus between make a difference of squares; with a plus it can't be factorised this way.
9. B. (2x + 7)(2x − 7)
4x² − 49 = (2x)² − 7² = (2x + 7)(2x − 7).
10. A. (2x + 3y)(2x − 3y)
4x² − 9y² = (2x)² − (3y)² = (2x + 3y)(2x − 3y).
11. False
Treat x + 10 as one term: (x + 10)² − 5² = (x + 10 + 5)(x + 10 − 5) = (x + 15)(x + 5).
12. B. (4x + 5y)(4x − 5y)
16x² − 25y² = (4x)² − (5y)² = (4x + 5y)(4x − 5y).

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 5, 2, 2
Take out 5 first: 5(x² − 4y²); then the difference of squares: 5(x + 2y)(x − 2y).
14. 440
57² − 53² = (57 + 53)(57 − 53) = 110 × 4 = 440.
15. A → C → B
Take out 5x first, write 36 as 6², then use the difference of squares.
16. C. x² + 9
A difference of squares needs a minus between two squares. x² + 9 has a plus: it can't be factorised this way.
17. False
Let A = x + 7 and B = x + 3: A² − B² = (A + B)(A − B) = (2x + 10)(4).
18. 2, 2
x² − 4 = x² − 2² = (x + 2)(x − 2).

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