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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 53 · 40 questions
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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.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
  2. 2.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 34 (c) 12 (d) 36
  3. 3.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94
  4. 4.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 3025 (d) 4225
  7. 7.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 8096 (d) 7744
  8. 8.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 5476 (d) 4366
  9. 9.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 116 (c) 196 (d) 36
  10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 539 (c) 289 (d) 409
  11. 11.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  12. 12.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 1156 (d) 884
  13. 13.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 76 (d) 100
  14. 14.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
  15. 15.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 67 (c) 81 (d) 109
  16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  17. 17.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
  18. 18.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 5 (d) 25
  19. 19.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 41 (c) 13 (d) 40
  20. 20.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 1 (c) 37 (d) 49
  21. 21.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 185 (d) 273
  22. 22.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1296 (c) 1306 (d) 1936
  23. 23.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1444 (d) 1604
  24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
  27. 27.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3364 (d) 3854
  28. 28.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81
  29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  30. 30.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1859 (c) 1609 (d) 1369
  31. 31.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 15 (d) 9
  32. 32.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
  33. 33.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 2 and 5 (d) 1 and 5
  34. 34.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 1 and 12 (d) 2 and 6
  35. 35.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 739 (d) 729
  36. 36.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  38. 38.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 17 (c) 21 (d) 33
  39. 39.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 81 (d) 25
  40. 40.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 121 (c) 61 (d) 41

Answer key

  1. 145
  2. 35
  3. 34
  4. 2 and 3
  5. (x + 6)(x + 7)
  6. 4225
  7. 7744
  8. 4356
  9. 36
  10. 529
  11. 256
  12. 676
  13. 52
  14. 8281
  15. 53
  16. (x + 7)(x + 9)
  17. 29
  18. 25
  19. 40
  20. 1
  21. 185
  22. 1296
  23. 1764
  24. (x + 8)(x + 9)
  25. (x + 2)(x + 7)
  26. 1444
  27. 3844
  28. 9
  29. 17
  30. 1849
  31. 8
  32. 15
  33. 2 and 4
  34. 2 and 6
  35. 729
  36. (x + 3)(x + 9)
  37. (x + 4)(x + 5)
  38. 17
  39. 25
  40. 41

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