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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 391 · 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 = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 81 (d) 45
  2. 2.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 27 (d) 11
  3. 3.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
  4. 4.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 9 (d) 8
  5. 5.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 196 (d) 106
  6. 6.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 4 (d) 144
  7. 7.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3721 (d) 3601
  8. 8.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
  9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
  10. 10.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 9 (d) 8
  11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
  12. 12.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 169 (d) 125
  13. 13.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 28 (d) 20
  14. 14.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 64 (c) 34 (d) 2
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  16. 16.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
  17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
  19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 48 (d) 13
  20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  21. 21.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 29 (d) 31
  22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 289 (d) 3
  23. 23.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 7 (d) 8
  24. 24.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
  26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
  27. 27.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 170 (c) 324 (d) 16
  28. 28.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 289
  29. 29.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 5 (c) 7 (d) 13
  30. 30.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  31. 31.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 1 and 13 (c) 3 and 15 (d) 5 and 9
  32. 32.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 12 (d) 13
  33. 33.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 4 (d) 58
  34. 34.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  35. 35.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2601 (c) 2411 (d) 2499
  36. 36.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 1 and 12 (d) 2 and 21
  37. 37.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 26 (c) 12 (d) 27
  38. 38.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 196 (c) 156 (d) 276
  39. 39.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3969 (d) 3979
  40. 40.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611

Answer key

  1. 45
  2. 28
  3. 7 and 8
  4. 9
  5. 106
  6. 4
  7. 3721
  8. 3025
  9. (x + 3)(x + 4)
  10. 9
  11. 72
  12. 81
  13. 20
  14. 4
  15. (x + 8)(x + 9)
  16. 9
  17. (x + 5)(x + 8)
  18. 1296
  19. 14
  20. (x + 3)(x + 6)
  21. 30
  22. 9
  23. 8
  24. (x + 4)(x + 6)
  25. (x + 2)(x + 5)
  26. 56
  27. 16
  28. 49
  29. 5
  30. (x + 6)(x + 9)
  31. 5 and 9
  32. 13
  33. 16
  34. (x + 6)(x + 7)
  35. 2401
  36. 6 and 7
  37. 27
  38. 116
  39. 3969
  40. 2601

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