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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 722 · 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 = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
  2. 2.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 130 (d) 2
  3. 3.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
  4. 4.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
  5. 5.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 15 (d) 17
  7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
  8. 8.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 127 (d) 85
  9. 9.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 196 (c) 100 (d) 292
  10. 10.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 125 (d) 147
  11. 11.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
  12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 111 (d) 101
  13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  14. 14.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 41 (d) 39
  15. 15.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 7 (c) 8 (d) 10
  16. 16.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
  17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  18. 18.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 100 (d) 10
  19. 19.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 137 (c) 225 (d) 313
  20. 20.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3364 (d) 3844
  21. 21.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 9 (c) 169 (d) 81
  22. 22.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 117 (d) 225
  23. 23.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 4096 (d) 3584
  24. 24.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)
  25. 25.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 19 (d) 1
  26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
  27. 27.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6734 (c) 6404 (d) 6084
  28. 28.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 884 (d) 676
  29. 29.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 5041 (d) 4761
  30. 30.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 73 (c) 5 (d) 121
  31. 31.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 2525 (d) 3025
  32. 32.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
  33. 33.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 16 (d) 4
  34. 34.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2916 (d) 2126
  35. 35.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
  36. 36.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 145 (c) 193 (d) 169
  37. 37.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 4225 (d) 5625
  38. 38.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 109 (d) 229
  39. 39.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 146 (d) 366
  40. 40.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 441 (d) 361

Answer key

  1. 25
  2. 4
  3. 9216
  4. 8 and 9
  5. (x + 6)(x + 9)
  6. 16
  7. 72
  8. 85
  9. 100
  10. 125
  11. 1225
  12. 101
  13. (x + 7)(x + 9)
  14. 40
  15. 7
  16. 576
  17. (x + 8)(x + 9)
  18. 100
  19. 137
  20. 3364
  21. 81
  22. 9
  23. 3136
  24. (x + 4)(x + 9)
  25. 1
  26. 7056
  27. 6724
  28. 676
  29. 4761
  30. 25
  31. 3025
  32. 74
  33. 16
  34. 2116
  35. 1444
  36. 145
  37. 4225
  38. 109
  39. 146
  40. 441

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