Algebraic expressions · Ages 13–14
Expanding perfect squares worksheet
Part of: Polynomials & identities
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Expanding perfect squares worksheet
Algebraic expressions · Ages 13–14
Numsprite
numsprite.com
Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.(x + 3)2 = x2 + x +
Fill in each blank
Answer:
2.(5x + 9)2 = x2 + x +
Fill in each blank
Answer:
3.(4x + 7)2 ≡ px2 + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
4.(5x + 2)2 ≡ px2 + qx + r, so p = , q = and r = .
Fill in each blank
Answer:
5.(5x + 5)2 = x2 + x +
Fill in each blank
Answer:
6.(4x + 5y)2 = x2 + xy + y2
Fill in each blank
Answer:
7.Work it out quickly with a perfect square: 212 =
Answer:
8.(2x + 4y)2 = x2 + xy + y2
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
9.(x − 8)2 = ?
- A. x2 − 64
- B. x2 + 64
- C. x2 − 8x + 64
- D. x2 − 16x + 64
Choose the right answer
Answer:
10.(x + 3)2 = x2 + 3x + 9 for every x.
True or false?
Answer:
11.(−x − 7)2 = ?
- A. x2 + 14x + 49
- B. x2 + 49
- C. −x2 − 14x − 49
- D. x2 − 14x + 49
Choose the right answer
Answer:
12.(−x − 5)2 = ?
- A. x2 + 10x + 25
- B. x2 − 10x + 25
- C. x2 + 25
- D. −x2 − 10x − 25
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.A square plaza has sides 69 m long. Working with (a − b)2, its area is m2.
Working:
Answer:
14.A square garden has sides x m long. With each side made 2 m longer, its area grows by (x + ) m2.
Fill in each blank
Working:
Answer:
15.Given x + y = 14 and xy = 48. Using identities, x2 + y2 = and (x − y)2 = .
Fill in each blank
Working:
Answer:
16.A square tile has sides (6x − 5) cm long. Its area is (x2 − x + ) cm2.
Fill in each blank
Working:
Answer:
17.Ka-yee says: (3x + 7)2 = 9x2 + 21x + 49.
True or false?
Working:
Answer:
18.Work out 982 with (a − b)2 ≡ a2 − 2ab + b2. Put the steps in order.
- A. 982 = (100 − 2)2
- B. = 9604
- C. = 1002 − 2 × 100 × 2 + 22
- D. = 10000 − 400 + 4
Write the letters in the right order
Working:
Answer:
Show the answers and working
Expanding perfect squares worksheet: Answers and working
Part A: Working out and filling in
1. 6, 9
First squared, x2; twice the product, 2 × x × 3 = 6x; last squared, 32 = 9.
2. 25, 90, 81
5x squared is 25x2; twice the product, 2 × 5x × 9 = 90x; 9 squared is 81.
3. 16, 56, 49
p = 42 = 16; q = 2 × 4 × (7) = 56, with the bracket's sign; r = (7)2 = 49: a square is never negative.
4. 25, 20, 4
p = 52 = 25; q = 2 × 5 × (2) = 20, with the bracket's sign; r = (2)2 = 4: a square is never negative.
5. 25, 50, 25
5x squared is 25x2; twice the product, 2 × 5x × 5 = 50x; 5 squared is 25.
6. 16, 40, 25
(4x)2 = 16x2; twice the product, 2 × 4x × 5y = 40xy; (5y)2 = 25y2.
7. 441
21 = 20 + 1: (20 + 1)2 = 400 + 40 + 1 = 441.
8. 4, 16, 16
(2x)2 = 4x2; twice the product, 2 × 2x × 4y = 16xy; (4y)2 = 16y2.
Part B: Choose, or true or false
9. D. x2 − 16x + 64
(a − b)2 ≡ a2 − 2ab + b2: x2 − 2 × x × 8 + 82 = x2 − 16x + 64.
10. False
The middle term is twice the product, 2 × x × 3 = 6x, so (x + 3)2 ≡ x2 + 6x + 9.
11. A. x2 + 14x + 49
(−x − 7)2 = [−(x + 7)]2 = (x + 7)2 = x2 + 14x + 49.
12. A. x2 + 10x + 25
(−x − 5)2 = [−(x + 5)]2 = (x + 5)2 = x2 + 10x + 25.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 4761
692 = (70 − 1)2 = 4900 − 140 + 1 = 4761.
14. 4, 4
The new area is (x + 2)2 = x2 + 4x + 4; minus the old x2, it grew by 4x + 4.
15. 100, 4
(x + y)2 = x2 + 2xy + y2, so x2 + y2 = 142 − 2 × 48 = 100; (x − y)2 = x2 + y2 − 2xy = 100 − 2 × 48 = 4.
16. 36, 60, 25
(6x − 5)2 = (6x)2 − 2 × 6x × 5 + 52 = 36x2 − 60x + 25.
17. False
The middle is twice the product: 2 × 3x × 7 = 42x, so (3x + 7)2 = 9x2 + 42x + 49.
18. A → C → D → B
Write 98 as 100 − 2, apply the identity, work out each term, then add and subtract.
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