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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 45 · 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.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 229 (d) 109
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  3. 3.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 45 (d) 3
  4. 4.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
  5. 5.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  6. 6.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 13 (d) 39
  7. 7.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 273 (d) 361
  8. 8.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 54 (d) 55
  9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 686 (d) 1156
  11. 11.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5625 (d) 5635
  13. 13.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
  14. 14.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 81 (c) 61 (d) 41
  15. 15.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 6425 (d) 5625
  16. 16.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
  18. 18.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 172 (d) 148
  19. 19.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 189 (d) 153
  20. 20.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 3 (d) 361
  21. 21.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 4 (d) 16
  22. 22.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 10 (d) 9
  23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  24. 24.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 256 (d) 352
  25. 25.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 4 (c) 2 (d) 20
  26. 26.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 16 (d) 256
  27. 27.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4899 (c) 4771 (d) 4761
  28. 28.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 148 (c) 292 (d) 196
  29. 29.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 9 (c) 81 (d) 225
  30. 30.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
  31. 31.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 12 (d) 35
  32. 32.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 58 (c) 79 (d) 142
  33. 33.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
  34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 57 (d) 55
  35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
  36. 36.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
  37. 37.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
  38. 38.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  39. 39.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 16 (d) 4
  40. 40.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)

Answer key

  1. 109
  2. (x + 8)(x + 9)
  3. 9
  4. 9
  5. (x + 2)(x + 5)
  6. 40
  7. 185
  8. 54
  9. (x + 2)(x + 6)
  10. 676
  11. (x + 2)(x + 9)
  12. 5625
  13. 3249
  14. 1
  15. 7225
  16. (x + 2)(x + 8)
  17. 1089
  18. 148
  19. 153
  20. 9
  21. 16
  22. 10
  23. (x + 6)(x + 9)
  24. 160
  25. 4
  26. 16
  27. 4761
  28. 100
  29. 81
  30. 1156
  31. 12
  32. 58
  33. 72
  34. 56
  35. 784
  36. 4
  37. 16
  38. 113
  39. 16
  40. (x + 2)(x + 3)

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