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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 744 · 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 = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 325 (d) 175
  2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
  3. 3.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 376 (d) 256
  4. 4.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 11 (d) 1 and 35
  5. 5.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5929 (d) 5939
  6. 6.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
  7. 7.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 153 (c) 225 (d) 9
  8. 8.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 1089 (d) 739
  9. 9.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7931 (d) 7921
  10. 10.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3136 (d) 3616
  11. 11.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 31 (d) 12
  12. 12.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 9035 (d) 7225
  13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
  14. 14.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 289 (d) 529
  15. 15.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  16. 16.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 61 (d) 81
  17. 17.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
  18. 18.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 30 (d) 31
  19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1849 (d) 1591
  21. 21.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 9 (d) 12
  22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
  23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  24. 24.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
  25. 25.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3599 (d) 3481
  26. 26.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 12 (d) 13
  27. 27.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  29. 29.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 49 (d) 25
  30. 30.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3601 (d) 3731
  31. 31.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9975 (d) 9025
  32. 32.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 8 (d) 10
  33. 33.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 129 (d) 249
  34. 34.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 4891 (d) 5329
  35. 35.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
  36. 36.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
  37. 37.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  38. 38.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  39. 39.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
  40. 40.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 97 (d) 133

Answer key

  1. 125
  2. 72
  3. 136
  4. 5 and 7
  5. 5929
  6. (x + 3)(x + 7)
  7. 81
  8. 729
  9. 7921
  10. 4096
  11. 32
  12. 9025
  13. (x + 2)(x + 9)
  14. 289
  15. (x + 2)(x + 8)
  16. 41
  17. 625
  18. 30
  19. (x + 4)(x + 7)
  20. 1369
  21. 8
  22. 484
  23. (x + 7)(x + 8)
  24. 13
  25. 3481
  26. 12
  27. 15
  28. (x + 8)(x + 9)
  29. 25
  30. 3721
  31. 9025
  32. 9
  33. 89
  34. 4489
  35. 13
  36. 10404
  37. 9
  38. (x + 4)(x + 8)
  39. 26
  40. 97

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