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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 873 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  2. 2.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 11
  3. 3.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 29 (c) 69 (d) 49
  4. 4.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  5. 5.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
  6. 6.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 121 (c) 3 (d) 9
  7. 7.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
  8. 8.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
  9. 9.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  10. 10.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55
  11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  12. 12.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 225 (d) 171
  13. 13.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 1 and 28 (c) 4 and 7 (d) 2 and 14
  14. 14.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  15. 15.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 113 (d) 1
  16. 16.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 8 (d) 15
  17. 17.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 3136 (d) 4096
  18. 18.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3025 (d) 3035
  19. 19.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 7056 (d) 6416
  20. 20.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 3 and 6 (d) 2 and 7
  21. 21.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
  22. 22.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
  23. 23.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 16 (d) 2 and 8
  24. 24.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1024 (d) 1034
  26. 26.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 10 (d) 13
  27. 27.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
  28. 28.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1596 (c) 1764 (d) 1444
  29. 29.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 17 (d) 15
  30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  31. 31.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
  32. 32.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8104 (d) 8464
  33. 33.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1774 (d) 1444
  34. 34.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 85 (d) 121
  35. 35.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4925 (c) 4225 (d) 5635
  36. 36.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 14 (d) 48
  37. 37.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 324 (c) 4 (d) 16
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 30 (d) 11
  39. 39.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)

Answer key

  1. (x + 8)(x + 9)
  2. 24
  3. 29
  4. 157
  5. 13
  6. 9
  7. 289
  8. 9
  9. (x + 4)(x + 8)
  10. 54
  11. (x + 2)(x + 4)
  12. 117
  13. 4 and 7
  14. (x + 4)(x + 5)
  15. 1
  16. 8
  17. 4096
  18. 3025
  19. 7056
  20. 2 and 7
  21. 65
  22. 36
  23. 2 and 8
  24. (x + 4)(x + 9)
  25. 1024
  26. 10
  27. 208
  28. 1444
  29. 16
  30. (x + 3)(x + 7)
  31. 17
  32. 8464
  33. 1764
  34. 85
  35. 5625
  36. 14
  37. 16
  38. 11
  39. (x + 2)(x + 6)
  40. (x + 3)(x + 6)

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