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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 123 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2714 (d) 2304
  2. 2.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 1 and 8 (d) 3 and 6
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  5. 5.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 4 (d) 400
  6. 6.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1296 (d) 1946
  7. 7.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 8 (d) 5
  8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
  9. 9.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 13 (d) 35
  10. 10.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
  11. 11.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
  12. 12.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 85 (d) 121
  13. 13.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 8 (d) 1 and 20
  14. 14.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 1 and 54 (d) 2 and 27
  15. 15.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 81 (c) 121 (d) 101
  16. 16.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1444 (d) 1774
  18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  20. 20.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 53 (c) 67 (d) 81
  21. 21.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 2 (d) 4
  22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  23. 23.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
  24. 24.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  26. 26.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 113 (d) 1
  27. 27.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 6375 (c) 5635 (d) 5625
  28. 28.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 6416 (d) 7066
  29. 29.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 9216 (d) 10816
  30. 30.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
  31. 31.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 49 (c) 61 (d) 37
  32. 32.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 409 (d) 289
  33. 33.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 169 (c) 49 (d) 109
  34. 34.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 153 (c) 297 (d) 225
  35. 35.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3136 (c) 4096 (d) 3616
  36. 36.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
  37. 37.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
  38. 38.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 333 (d) 441
  39. 39.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 9 (d) 65
  40. 40.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)

Answer key

  1. 2704
  2. 3 and 6
  3. (x + 6)(x + 8)
  4. (x + 6)(x + 7)
  5. 16
  6. 1936
  7. 6
  8. 11
  9. 12
  10. 625
  11. (x + 3)(x + 7)
  12. 49
  13. 4 and 5
  14. 6 and 9
  15. 81
  16. (x + 3)(x + 4)
  17. 1764
  18. 17
  19. 72
  20. 53
  21. 4
  22. (x + 6)(x + 9)
  23. (x + 4)(x + 7)
  24. (x + 2)(x + 6)
  25. (x + 7)(x + 8)
  26. 1
  27. 5625
  28. 7056
  29. 10816
  30. 8
  31. 37
  32. 529
  33. 49
  34. 153
  35. 4096
  36. (x + 5)(x + 7)
  37. 100
  38. 225
  39. 9
  40. (x + 4)(x + 6)

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