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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 365 · 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.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  2. 2.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 109 (d) 74
  3. 3.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 45 (c) 14 (d) 44
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
  6. 6.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  7. 7.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 149 (d) 289
  8. 8.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 273 (c) 537 (d) 361
  9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2601 (d) 2501
  11. 11.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 26 (c) 46 (d) 31
  12. 12.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 65 (d) 121
  13. 13.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
  14. 14.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 133 (d) 97
  15. 15.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
  16. 16.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
  17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
  18. 18.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 5476 (c) 4356 (d) 4884
  19. 19.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
  20. 20.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 81 (c) 97 (d) 73
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  22. 22.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1531 (d) 1521
  23. 23.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 904 (d) 1034
  24. 24.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
  25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1296 (c) 1946 (d) 1616
  26. 26.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
  27. 27.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 24 (d) 10
  28. 28.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 11 (d) 1 and 35
  29. 29.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
  30. 30.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 109 (d) 139
  31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  32. 32.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  33. 33.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 253 (d) 169
  34. 34.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 16 (d) 63
  35. 35.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  37. 37.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)
  38. 38.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625
  40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18

Answer key

  1. (x + 4)(x + 5)
  2. 74
  3. 45
  4. (x + 8)(x + 9)
  5. 15
  6. (x + 2)(x + 5)
  7. 149
  8. 185
  9. (x + 3)(x + 6)
  10. 2601
  11. 26
  12. 9
  13. 3249
  14. 97
  15. 961
  16. 256
  17. 2704
  18. 4356
  19. 9
  20. 65
  21. 181
  22. 1521
  23. 1024
  24. 5776
  25. 1936
  26. 4
  27. 11
  28. 5 and 7
  29. 484
  30. 109
  31. (x + 3)(x + 4)
  32. (x + 7)(x + 9)
  33. 85
  34. 63
  35. 72
  36. (x + 4)(x + 8)
  37. (x + 7)(x + 8)
  38. (x + 6)(x + 9)
  39. 5625
  40. 17

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