Algebraic expressions · Ages 13–14

Expanding the difference of two squares worksheet

Part of: Polynomials & identities

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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.

1.(x + 6)(x − 6) = x² −
Answer:
2.(x + 2)(x − 2) = x² −
Answer:
3.(4x + 5y)(4x − 5y) = x² − y²
Fill in each blank
Answer:
4.Work it out quickly with the difference of two squares: 25 × 35 =
Answer:
5.(4x + 4y)(4x − 4y) = x² − y²
Fill in each blank
Answer:
6.(x + 4)(x − 4) + (x − 1)² ≡ 2x² + px + q, so p = and q = .
Fill in each blank
Answer:
7.(x + 2)(x − 2) + (x − 7)² ≡ 2x² + px + q, so p = and q = .
Fill in each blank
Answer:

8.(5x + 4)(5x − 4) = ?
- A. 5x² − 16
- B. 25x² + 16
- C. 25x² − 16
- D. 25x² − 40x − 16
Choose the right answer
Answer:
9.(4x + 5)(4x − 5) = ?
- A. 16x² − 40x − 25
- B. 4x² − 25
- C. 16x² − 25
- D. 16x² + 25
Choose the right answer
Answer:
10.(12 + x)(x − 12) = x² − 24 for every x.
True or false?
Answer:
11.(10 + x)(x − 10) = x² − 100 for every x.
True or false?
Answer:
12.(2x + 1)(1 − 2x) = ?
- A. 4x² − 1
- B. 1 − 4x²
- C. 1 − 4x² + 4x
- D. 1 + 4x²
Choose the right answer
Answer:

13.Ka-lok says: 27 × 33 = 30² + 3².
True or false?
Working:
Answer:
14.A storeroom holds 34 boxes of 26 items each. Using the difference of two squares, there are items in all.
Working:
Answer:
15.Prove (x + 7)(x − 7) ≡ x² − 49. Put the steps in order.
- A. = right side
- B. = x² − 49
- C. Left side = x² − 7x + 7x − 49
Write the letters in the right order
Working:
Answer:
16.Simplify (x + 10)(x − 10) − (x + 4)(x − 4). The result is .
Working:
Answer:
17.A square has sides x cm long. It is made into a rectangle 6 cm longer and 6 cm narrower. The new area is (x² − ) cm², which is cm² less than before.
Fill in each blank
Working:
Answer:
18.A rectangle is (2x + 7) m long and (2x − 7) m wide. How many m² less is its area than that of a square of side 2x m?
- A. 14
- B. 7
- C. 49
- D. 4x² − 49
Choose the right answer
Working:
Answer:

Show the answers and working

Expanding the difference of two squares worksheet: Answers and working

1. 36
A sum times a difference is a difference of squares: −6x and +6x cancel, leaving x² − 6² = x² − 36.
2. 4
A sum times a difference is a difference of squares: −2x and +2x cancel, leaving x² − 2² = x² − 4.
3. 16, 25
A difference of squares: (4x)² − (5y)² = 16x² − 25y².
4. 875
25 × 35 = (30 − 5)(30 + 5) = 30² − 5² = 900 − 25 = 875.
5. 16, 16
A difference of squares: (4x)² − (4y)² = 16x² − 16y².
6. −2, −15
x² − 16 plus x² + 2(−1)x + 1: p = 2 × (−1) = −2, q = 1 − 16 = −15.
7. −14, 45
x² − 4 plus x² + 2(−7)x + 49: p = 2 × (−7) = −14, q = 49 − 4 = 45.

8. C. 25x² − 16
(a + b)(a − b) ≡ a² − b²: (5x)² − 4² = 25x² − 16.
9. C. 16x² − 25
(a + b)(a − b) ≡ a² − b²: (4x)² − 5² = 16x² − 25.
10. False
(12 + x) is (x + 12), so it is a sum times a difference: x² − 12² = x² − 144.
11. True
(10 + x) is (x + 10), so it is a sum times a difference: x² − 10² = x² − 100.
12. B. 1 − 4x²
Reorder first: (1 + 2x)(1 − 2x) = 1² − (2x)² = 1 − 4x².

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

13. False
27 × 33 = (30 − 3)(30 + 3), a sum times a difference: 30² − 3².
14. 884
34 × 26 = (30 + 4)(30 − 4) = 900 − 16 = 884.
15. C → B → A
Multiply term by term; the middle −7x and +7x cancel, and what is left is the right side.
16. −84
(x² − 100) − (x² − 16) = x² − 100 − x² + 16 = −84.
17. 36, 36
(x + 6)(x − 6) = x² − 36, which is 36 less than the square's x².
18. C. 49
The rectangle's area = (2x + 7)(2x − 7) = 4x² − 49; the square's is 4x²: 49 less.

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