Algebraic expressions · Ages 13–14
Factorising perfect squares worksheet
Part of: Factorisation
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Factorising perfect squares worksheet
Algebraic expressions · Ages 13–14
Numsprite
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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.x2 + 10x + 25 = (x + )2
Answer:
2.x2 + 4x + 4 = (x + )2
Answer:
3.6x2 − 108x + 486 = (x − )2
Fill in each blank
Answer:
4.4x2 + 28x + 49 = (x + )2
Fill in each blank
Answer:
5.4x2 + kx + 25 is a perfect square and k is positive, so k = .
Answer:
6.x2 + 20x + k is a perfect square, so k = .
Answer:
7.x2 + 12x + k is a perfect square, so k = .
Answer:
Part B: Choose, or true or false (2 marks each)
8.Factorise x2 − 14x + 49. What do you get?
- A. (x − 14)2
- B. (x + 7)2
- C. (x − 7)2
- D. (x + 7)(x − 7)
Choose the right answer
Answer:
9.x2 + 12x + 36 is a perfect square.
True or false?
Answer:
10.x2 + 14x + 49 is a perfect square.
True or false?
Answer:
11.Factorise 16x2 − 24xy + 9y2. What do you get?
- A. (4x − 3y)2
- B. (3x − 4y)2
- C. (4x + 3y)2
- D. (4x + 3y)(4x − 3y)
Choose the right answer
Answer:
12.Factorise 9x2 − 30xy + 25y2. What do you get?
- A. (3x + 5y)(3x − 5y)
- B. (3x − 5y)2
- C. (5x − 3y)2
- D. (3x + 5y)2
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.A square rug has an area of (x2 + 4x + 4) m2. Its side is (x + ) m long; when x = 13, the side is m.
Fill in each blank
Working:
Answer:
14.x2 − kx + 25 is a perfect square and k is positive. k =
Working:
Answer:
15.Factorise 64x2 + 224x + 196 completely: 64x2 + 224x + 196 = (x + )2.
Fill in each blank
Working:
Answer:
16.Ka-yee says: 302 − 2 × 30 × 3 + 32 = 891.
True or false?
Working:
Answer:
17.Use a perfect square: 182 + 2 × 18 × 62 + 622 =
Working:
Answer:
18.Factorise 25x2 − 90x + 81. Put the steps in order.
- A. 25x2 − 90x + 81 = (5x − 9)2
- B. Middle: 90x = 2 × 5x × 9
- C. First and last: 25x2 = (5x)2, 81 = 92
Write the letters in the right order
Working:
Answer:
Show the answers and working
Factorising perfect squares worksheet: Answers and working
Part A: Working out and filling in
1. 5
25 = 52 and the middle 10x = 2 × x × 5, so it is (x + 5)2.
2. 2
4 = 22 and the middle 4x = 2 × x × 2, so it is (x + 2)2.
3. 6, 9
Take out the common factor 6: 6(x2 − 18x + 81); inside is a perfect square: 6(x − 9)2.
4. 2, 7
4x2 = (2x)2, 49 = 72 and the middle 28x = 2 × 2x × 7, so it is (2x + 7)2.
5. 20
4x2 = (2x)2 and 25 = 52; the middle is twice the product, 2 × 2x × 5 = 20x, so k = 20.
6. 100
The middle 20x = 2 × x × 10, so the last term is 102 = 100: x2 + 20x + 100 = (x + 10)2.
7. 36
The middle 12x = 2 × x × 6, so the last term is 62 = 36: x2 + 12x + 36 = (x + 6)2.
Part B: Choose, or true or false
8. C. (x − 7)2
x2 and 49 = 72 are squares and the middle 14x = 2 × x × 7, with a minus: (x − 7)2.
9. True
36 = 62, so a perfect square's middle term would be 2 × x × 6 = 12x.
10. True
49 = 72, so a perfect square's middle term would be 2 × x × 7 = 14x.
11. A. (4x − 3y)2
16x2 = (4x)2, 9y2 = (3y)2 and the middle 24xy = 2 × 4x × 3y, with a minus: (4x − 3y)2.
12. B. (3x − 5y)2
9x2 = (3x)2, 25y2 = (5y)2 and the middle 30xy = 2 × 3x × 5y, with a minus: (3x − 5y)2.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 2, 15
x2 + 4x + 4 = (x + 2)2, so the side is x + 2; when x = 13, 13 + 2 = 15.
14. 10
25 = 52, so the middle is 2 × x × 5 = 10x: x2 − 10x + 25 = (x − 5)2 and k = 10.
15. 4, 4, 7
Take out 4; inside, 16x2 + 56x + 49 is (4x + 7)2.
16. False
a2 − 2ab + b2 = (a − b)2: (30 − 3)2 = 272 = 729.
17. 6400
a2 + 2ab + b2 = (a + b)2: (18 + 62)2 = 802 = 6400.
18. C → B → A
Spot the squares at the ends, check the middle is twice their product, then write (5x − 9)2 with the middle's minus inside.
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