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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 71 · 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 + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  2. 2.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 117 (d) 171
  3. 3.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 676 (d) 1166
  4. 4.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3025 (d) 3575
  5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
  6. 6.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 9 (d) 20
  7. 7.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 10 (d) 12
  8. 8.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 97 (c) 5 (d) 169
  9. 9.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 196
  10. 10.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
  11. 11.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 2 (c) 164 (d) 4
  12. 12.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 10816 (d) 9216
  14. 14.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
  15. 15.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
  16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 875 (c) 1225 (d) 635
  18. 18.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 169 (c) 109 (d) 229
  19. 19.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 361 (d) 273
  20. 20.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 196 (d) 244
  21. 21.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
  22. 22.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 11
  23. 23.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 9 (d) 24
  25. 25.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2496 (d) 2304
  26. 26.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 416 (d) 256
  27. 27.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68
  28. 28.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  29. 29.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 17 (c) 21 (d) 33
  30. 30.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2501 (d) 2601
  31. 31.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 121 (d) 61
  32. 32.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 256 (d) 196
  33. 33.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 39 (d) 13
  34. 34.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 100 (c) 82 (d) 64
  35. 35.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 15 (c) 17 (d) 10
  36. 36.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 1 and 15 (d) 3 and 21
  37. 37.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
  38. 38.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 961 (d) 841
  39. 39.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
  40. 40.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 149 (d) 219

Answer key

  1. (x + 4)(x + 7)
  2. 117
  3. 1156
  4. 3025
  5. (x + 5)(x + 9)
  6. 9
  7. 11
  8. 25
  9. 4
  10. 25
  11. 4
  12. (x + 4)(x + 9)
  13. 9216
  14. 729
  15. 16
  16. (x + 6)(x + 9)
  17. 625
  18. 109
  19. 185
  20. 148
  21. 36
  22. 24
  23. 6241
  24. 10
  25. 2304
  26. 576
  27. 68
  28. 225
  29. 17
  30. 2601
  31. 61
  32. 136
  33. 40
  34. 64
  35. 16
  36. 7 and 9
  37. 1
  38. 841
  39. 73
  40. 149

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