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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 204 · 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.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 169 (d) 11
  2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
  5. 5.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
  6. 6.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 9 (d) 1
  7. 7.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4106 (d) 4096
  9. 9.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
  10. 10.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 31 (d) 26
  11. 11.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 8291 (d) 7921
  12. 12.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  13. 13.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 400 (d) 208
  14. 14.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 657 (d) 441
  15. 15.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  16. 16.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
  17. 17.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 53 (c) 5 (d) 81
  18. 18.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 6889 (d) 5929
  19. 19.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 193 (d) 25
  20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  21. 21.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 8 (d) 9
  22. 22.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
  23. 23.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 10 (d) 11
  24. 24.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 7406 (c) 8836 (d) 8084
  25. 25.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 81 (d) 121
  26. 26.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 289 (d) 169
  27. 27.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 54 (d) 15
  28. 28.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 6425 (d) 7225
  29. 29.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1604 (d) 1764
  30. 30.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 7
  31. 31.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3844 (c) 3854 (d) 3364
  32. 32.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  33. 33.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 529 (c) 391 (d) 299
  34. 34.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 30 (d) 11
  35. 35.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  36. 36.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 116 (c) 196 (d) 156
  37. 37.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
  38. 38.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 916 (d) 1156
  40. 40.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 4 (d) 2

Answer key

  1. 121
  2. (x + 2)(x + 4)
  3. (x + 8)(x + 9)
  4. 164
  5. 5 and 6
  6. 1
  7. (x + 3)(x + 9)
  8. 4096
  9. 9
  10. 26
  11. 8281
  12. (x + 4)(x + 8)
  13. 208
  14. 225
  15. (x + 2)(x + 5)
  16. 8 and 9
  17. 25
  18. 5929
  19. 25
  20. (x + 2)(x + 3)
  21. 9
  22. 2916
  23. 11
  24. 7396
  25. 41
  26. 49
  27. 15
  28. 7225
  29. 1764
  30. 2 and 6
  31. 3844
  32. 145
  33. 289
  34. 30
  35. (x + 3)(x + 7)
  36. 116
  37. 1
  38. (x + 6)(x + 8)
  39. 1156
  40. 4

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