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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 218 · 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 = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
  2. 2.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 193 (d) 529
  3. 3.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 256 (c) 196 (d) 376
  4. 4.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 64 (c) 196 (d) 130
  5. 5.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2304 (c) 2704 (d) 2496
  6. 6.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 2025 (d) 1225
  7. 7.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)
  8. 8.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)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 576 (d) 586
  10. 10.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 64 (d) 62
  11. 11.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 10 (d) 11
  12. 12.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
  13. 13.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1521 (d) 1681
  14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 185 (d) 537
  15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  16. 16.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 97 (c) 169 (d) 25
  17. 17.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 12 (d) 32
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  19. 19.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 196 (d) 116
  20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 81 (d) 61
  21. 21.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
  22. 22.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 121 (c) 3 (d) 65
  23. 23.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 12 (d) 33
  24. 24.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 85 (c) 7 (d) 121
  25. 25.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 217 (d) 289
  26. 26.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 10 (d) 22
  27. 27.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 9 (c) 4 and 6 (d) 1 and 24
  28. 28.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)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6404 (d) 6734
  30. 30.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9226 (d) 9984
  31. 31.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 196 (c) 106 (d) 16
  32. 32.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
  33. 33.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 4634 (d) 5184
  34. 34.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 144 (d) 214
  35. 35.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 16 (d) 136
  36. 36.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
  37. 37.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 14 (d) 13
  38. 38.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  39. 39.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
  40. 40.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 19 (d) 18

Answer key

  1. 170
  2. 193
  3. 136
  4. 64
  5. 2304
  6. 1225
  7. (x + 7)(x + 9)
  8. (x + 3)(x + 6)
  9. 576
  10. 63
  11. 11
  12. 7
  13. 1521
  14. 185
  15. (x + 5)(x + 8)
  16. 25
  17. 12
  18. 15
  19. 116
  20. 1
  21. 7 and 9
  22. 9
  23. 32
  24. 49
  25. 1
  26. 10
  27. 4 and 6
  28. (x + 5)(x + 6)
  29. 6724
  30. 9216
  31. 16
  32. 9
  33. 4624
  34. 74
  35. 16
  36. 5184
  37. 14
  38. 8 and 9
  39. 4225
  40. 18

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