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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 126 · 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.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
  2. 2.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6
  3. 3.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
  4. 4.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 94 (d) 34
  5. 5.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 8 (d) 12
  6. 6.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
  7. 7.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 81 (d) 25
  8. 8.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
  9. 9.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 6 (c) 90 (d) 36
  10. 10.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
  11. 11.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
  12. 12.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 4106 (c) 3136 (d) 3616
  13. 13.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
  14. 14.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 225 (c) 49 (d) 7
  15. 15.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
  16. 16.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 1 and 11 (d) 2 and 10
  17. 17.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 19 (c) 9 (d) 21
  18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  19. 19.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
  20. 20.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 208 (c) 144 (d) 112
  21. 21.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 180 (c) 324 (d) 6
  22. 22.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)
  23. 23.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 52 (d) 100
  24. 24.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)
  25. 25.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 11 (d) 10
  26. 26.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3844 (d) 3364
  27. 27.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 145 (c) 169 (d) 193
  28. 28.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 19 (d) 25
  29. 29.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 598 (d) 301
  30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 149 (d) 9
  32. 32.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  33. 33.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4624 (d) 4896
  34. 34.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 28 (d) 10
  35. 35.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
  36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4356 (c) 4884 (d) 4366
  38. 38.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 14 (d) 13
  39. 39.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 156 (c) 196 (d) 276
  40. 40.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4875 (d) 4225

Answer key

  1. 8 and 9
  2. 3 and 5
  3. 2601
  4. 34
  5. 12
  6. 6 and 9
  7. 25
  8. 26
  9. 36
  10. 3025
  11. 35
  12. 4096
  13. 45
  14. 49
  15. 63
  16. 5 and 7
  17. 20
  18. (x + 8)(x + 9)
  19. 13
  20. 80
  21. 36
  22. (x + 2)(x + 6)
  23. 52
  24. (x + 7)(x + 9)
  25. 11
  26. 3364
  27. 145
  28. 13
  29. 202
  30. (x + 5)(x + 9)
  31. 9
  32. 1681
  33. 4624
  34. 11
  35. 6
  36. (x + 2)(x + 7)
  37. 4356
  38. 14
  39. 116
  40. 4225

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