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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 523 · 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.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 539 (d) 289
  2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  3. 3.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  5. 5.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 149 (d) 219
  6. 6.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 1 and 5 (d) 2 and 5
  7. 7.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 225 (c) 117 (d) 9
  8. 8.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2475 (d) 2025
  9. 9.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  10. 10.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 17 (d) 15
  11. 11.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 5929 (d) 6391
  12. 12.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2516 (c) 2926 (d) 2116
  13. 13.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 7744 (d) 8104
  14. 14.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 289 (c) 529 (d) 391
  15. 15.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 5 and 6 (d) 1 and 30
  16. 16.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
  17. 17.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 196 (c) 4 (d) 100
  18. 18.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 8 (c) 130 (d) 64
  19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  20. 20.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 11 (c) 23 (d) 24
  21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
  22. 22.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 5 (d) 1
  23. 23.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 100 (d) 79
  24. 24.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 2 (d) 4
  25. 25.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)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
  27. 27.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 25 (d) 5
  28. 28.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
  29. 29.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
  30. 30.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
  31. 31.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  33. 33.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 16 (c) 14 (d) 8
  34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 4624 (d) 5184
  37. 37.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  38. 38.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 196 (c) 130 (d) 262
  39. 39.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
  40. 40.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 784 (d) 896

Answer key

  1. 529
  2. 72
  3. 20
  4. (x + 8)(x + 9)
  5. 149
  6. 2 and 4
  7. 9
  8. 2025
  9. 8
  10. 16
  11. 5929
  12. 2916
  13. 8464
  14. 289
  15. 5 and 6
  16. 14
  17. 4
  18. 64
  19. (x + 5)(x + 6)
  20. 24
  21. 145
  22. 1
  23. 58
  24. 4
  25. (x + 2)(x + 9)
  26. 5625
  27. 25
  28. 16
  29. 676
  30. 18
  31. 8836
  32. (x + 5)(x + 7)
  33. 15
  34. (x + 7)(x + 8)
  35. (x + 6)(x + 7)
  36. 4624
  37. 157
  38. 130
  39. 68
  40. 784

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