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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 152 · 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.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2496 (c) 2314 (d) 2704
  2. 2.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25
  3. 3.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 1 and 13 (d) 3 and 15
  4. 4.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1444 (d) 1764
  5. 5.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 12 (d) 30
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  7. 7.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 73 (d) 121
  8. 8.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 36 (c) 6 (d) 68
  9. 9.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 11 (d) 30
  10. 10.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 89 (d) 3
  11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3481 (d) 3601
  12. 12.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 625 (d) 225
  13. 13.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 61 (d) 41
  14. 14.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 9025 (d) 10025
  15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 181 (d) 271
  16. 16.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  17. 17.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 784 (c) 794 (d) 1024
  18. 18.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 15 (c) 8 (d) 16
  19. 19.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9
  20. 20.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 8836 (d) 7396
  21. 21.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 425 (d) 635
  22. 22.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
  23. 23.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 9 (d) 7
  24. 24.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 136 (c) 376 (d) 196
  25. 25.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 8 (c) 3 and 6 (d) 1 and 14
  26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  28. 28.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 13 (c) 22 (d) 16
  29. 29.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
  30. 30.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
  31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9035 (d) 9025
  33. 33.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
  34. 34.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  35. 35.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 63 (d) 117
  37. 37.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 409 (d) 539
  38. 38.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2601 (d) 2501
  39. 39.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 40 (c) 13 (d) 14
  40. 40.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 7235 (d) 9025

Answer key

  1. 2304
  2. 25
  3. 5 and 9
  4. 1764
  5. 11
  6. (x + 8)(x + 9)
  7. 25
  8. 36
  9. 30
  10. 9
  11. 3721
  12. 225
  13. 41
  14. 11025
  15. 1
  16. 185
  17. 784
  18. 15
  19. 8 and 9
  20. 7396
  21. 625
  22. 82
  23. 8
  24. 136
  25. 2 and 7
  26. 15
  27. 17
  28. 10
  29. 25
  30. 1
  31. (x + 2)(x + 7)
  32. 9025
  33. 42
  34. (x + 3)(x + 6)
  35. 180
  36. 45
  37. 529
  38. 2601
  39. 13
  40. 7225

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