Algebraic expressions · Ages 13–14
Identities and equations worksheet
Part of: Polynomials & identities
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Identities and equations 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.For 3(x − 5) = 3x − k to be an identity, k = .
Answer:
2.For 3(x − 3) = 3x − k to be an identity, k = .
Answer:
3.Is (x + 2)2 = x2 + 4 an identity? Put in x = 3: the left side is and the right side is .
Fill in each blank
Answer:
4.Is (x + 6)2 = x2 + 36 an identity? Put in x = 3: the left side is and the right side is .
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
5.Which of these is an identity?
- A. 8x + 9 = 81
- B. 8(x + 9) = 8x + 9
- C. 8(x + 9) = 8x + 72
- D. (x + 9)2 = x2 + 81
Choose the right answer
Answer:
6.5(x + 2) = 5x + 2 is an identity.
True or false?
Answer:
7.5(x + 5) = 5x + 5 is an identity.
True or false?
Answer:
8.(x − 7)(x + 7) = x2 − 49 is an identity.
True or false?
Answer:
9.(x − 10)(x + 10) = x2 − 20 is an identity.
True or false?
Answer:
10.6(x + 4) + 3(x − 3) = 9x + 15 is an identity.
True or false?
Answer:
11.Which kind is the equality 6(x + 6) = 3x + 36?
- A. An equation with no solution
- B. An equation with one solution
- C. An equation with two solutions
- D. An identity: true for every x
Choose the right answer
Answer:
12.Which of these is not an identity?
- A. (x − 8)(x + 8) = x2 − 64
- B. (x + 6)2 = x2 + 12x + 36
- C. (x − 6)2 = x2 − 36
- D. 6(x − 8) + 3 = 6x − 45
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.The equality 5(x + 4) = 2x + 17 is not an identity: it holds only when x = .
Working:
Answer:
14.Prove (x + 10)2 − (x − 10)2 ≡ 40x. Left side = (x2 + x + 100) − (x2 − x + 100) = x = right side.
Fill in each blank
Working:
Answer:
15.Ka-yan says: 'When x = 8, both sides of 6(x + 4) = 5x + 32 are equal, so it is an identity.' Is that right?
- A. No: it has no solution
- B. Yes: the x coefficients match
- C. No: it is an equation, true only when x = 8
- D. Yes: the sides agree when the value is put in
Choose the right answer
Working:
Answer:
16.The equality 9(x − 4) + 16 = 9x − 52 holds for every x.
True or false?
Working:
Answer:
17.Which of these is an identity?
- A. (x − 2)2 = (x + 2)2 − 8x
- B. 9(x − 2) = 9x − 2
- C. (x − 2)2 = x2 − 4
- D. (x + 2)(x − 2) = (x − 2)2
Choose the right answer
Working:
Answer:
18.Prove (x + 9)(x + 2) − 18 ≡ x(x + 11). Put the steps in order.
- A. = x(x + 11)
- B. = x2 + 11x
- C. Left side = x2 + 2x + 9x + 18 − 18
- D. = right side
Write the letters in the right order
Working:
Answer:
Show the answers and working
Identities and equations worksheet: Answers and working
Part A: Working out and filling in
1. 15
The left expands to 3x − 15. Both sides must match exactly, so k = 15.
2. 9
The left expands to 3x − 9. Both sides must match exactly, so k = 9.
3. 25, 13
Left: (3 + 2)2 = 25; right: 32 + 4 = 13. One value where they differ is enough: it is not an identity. The identity is (x + 2)2 ≡ x2 + 4x + 4.
4. 81, 45
Left: (3 + 6)2 = 81; right: 32 + 36 = 45. One value where they differ is enough: it is not an identity. The identity is (x + 6)2 ≡ x2 + 12x + 36.
Part B: Choose, or true or false
5. C. 8(x + 9) = 8x + 72
An identity simplifies to exactly the same on both sides: 8(x + 9) = 8x + 72, true for every x.
6. False
Expand the left: 5(x + 2) = 5x + 10. Only when both sides simplify to exactly the same is it an identity.
7. False
Expand the left: 5(x + 5) = 5x + 25. Only when both sides simplify to exactly the same is it an identity.
8. True
Expand the left: x2 + 7x − 7x − 49 = x2 − 49. It is an identity only if that is exactly the right side.
9. False
Expand the left: x2 + 10x − 10x − 100 = x2 − 100. It is an identity only if that is exactly the right side.
10. True
Expand both brackets, every term multiplied: 6x + 24 + 3x − 9 = 9x + 15. Only exactly that on the right makes an identity.
11. B. An equation with one solution
The left expands to 6x + 36. Against 3x + 36: exactly the same, an identity; same x coefficient but different constants, no solution; different x coefficients, one solution.
12. C. (x − 6)2 = x2 − 36
(x − 6)2 = (x − 6)(x − 6) = x2 − 12x + 36, not x2 − 36. Putting in x = 0 shows it too: 36 on the left, −36 on the right.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. −1
Expand: 5x + 20 = 2x + 17. The x coefficients differ, so it is an equation: (5 − 2)x = 17 − 20, x = −1.
14. 20, 20, 40
Expanding both squares, x2 and 100 cancel, leaving 20x + 20x = 40x, the right side: so it is an identity.
15. C. No: it is an equation, true only when x = 8
The left expands to 6x + 24; its x coefficient differs from 5x + 32's, so there is one solution, x = 8: an equation, not an identity.
16. False
Expand the left: 9x − 36 + 16 = 9x − 20. Only if the right's constant is −20 is it an identity.
17. A. (x − 2)2 = (x + 2)2 − 8x
(x + 2)2 − 8x = x2 + 4x + 4 − 8x = x2 − 4x + 4 = (x − 2)2: exactly the same.
18. C → B → A → D
Start from the left, expand and simplify, and arrive at the right: the two sides agree for every x.
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