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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 789 · 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 93²? (a) 7569 (b) 8659 (c) 8649 (d) 8109
  2. 2.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 20 (d) 1 and 8
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 7 (c) 9 (d) 15
  4. 4.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 478 (d) 170
  5. 5.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 625 (c) 635 (d) 1225
  6. 6.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9801 (c) 9811 (d) 10201
  7. 7.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  9. 9.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 54 (d) 53
  10. 10.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  11. 11.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  12. 12.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 7396 (c) 8116 (d) 8846
  13. 13.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 6 (d) 116
  14. 14.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 4 (d) 196
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  16. 16.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 399 (c) 361 (d) 371
  17. 17.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 17 (c) 15 (d) 10
  18. 18.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 5625 (d) 7235
  19. 19.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 49 (c) 137 (d) 225
  20. 20.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 127 (d) 253
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6409 (d) 6889
  23. 23.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 4 and 8 (d) 1 and 32
  24. 24.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 144 (c) 100 (d) 122
  25. 25.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
  26. 26.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 4 (d) 16
  27. 27.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
  28. 28.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 121 (d) 85
  29. 29.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
  30. 30.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
  31. 31.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  32. 32.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4096 (d) 4106
  34. 34.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 16 (d) 2
  35. 35.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 225 (c) 297 (d) 153
  36. 36.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 4 (c) 1 and 8 (d) 2 and 5
  37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  38. 38.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  39. 39.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416

Answer key

  1. 8649
  2. 4 and 5
  3. 8
  4. 170
  5. 625
  6. 9801
  7. 185
  8. (x + 5)(x + 6)
  9. 54
  10. (x + 4)(x + 8)
  11. 16
  12. 8836
  13. 36
  14. 16
  15. 17
  16. 361
  17. 16
  18. 7225
  19. 49
  20. 85
  21. 181
  22. 6889
  23. 4 and 8
  24. 100
  25. 7921
  26. 16
  27. 10
  28. 85
  29. 289
  30. 202
  31. (x + 6)(x + 7)
  32. (x + 3)(x + 9)
  33. 4096
  34. 4
  35. 153
  36. 2 and 4
  37. (x + 2)(x + 7)
  38. 15
  39. (x + 8)(x + 9)
  40. 576

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