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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 356 · 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.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 9 (d) 29
  2. 2.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
  3. 3.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 35 (d) 37
  4. 4.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 12 (d) 11
  5. 5.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 433 (d) 145
  6. 6.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 36 (d) 52
  7. 7.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 1 and 7 (c) 3 and 4 (d) 2 and 6
  8. 8.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  10. 10.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 25 (d) 53
  11. 11.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 61 (c) 91 (d) 121
  12. 12.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 9409 (d) 10009
  13. 13.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
  14. 14.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 118 (c) 100 (d) 82
  15. 15.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1454 (d) 1764
  16. 16.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 46 (c) 14 (d) 45
  17. 17.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  18. 18.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
  19. 19.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
  20. 20.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 6 (d) 116
  21. 21.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2525 (c) 3025 (d) 2025
  22. 22.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 97 (c) 65 (d) 81
  23. 23.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 18 (d) 9
  24. 24.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 52 (d) 4
  25. 25.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  28. 28.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
  29. 29.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14
  30. 30.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  32. 32.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 4 (d) 144
  33. 33.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 144 (d) 124
  34. 34.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1225 (d) 1575
  35. 35.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 4875 (d) 5625
  36. 36.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5776 (c) 6384 (d) 5786
  37. 37.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  38. 38.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18
  40. 40.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9

Answer key

  1. 9
  2. 5929
  3. 36
  4. 12
  5. 145
  6. 20
  7. 2 and 6
  8. 8 and 9
  9. (x + 7)(x + 8)
  10. 25
  11. 1
  12. 10609
  13. 11
  14. 82
  15. 1444
  16. 45
  17. 1
  18. 12
  19. 961
  20. 36
  21. 3025
  22. 65
  23. 9
  24. 4
  25. (x + 5)(x + 9)
  26. (x + 4)(x + 7)
  27. (x + 4)(x + 8)
  28. (x + 2)(x + 6)
  29. 14
  30. (x + 2)(x + 3)
  31. 1
  32. 4
  33. 104
  34. 1225
  35. 4225
  36. 5776
  37. (x + 2)(x + 9)
  38. 164
  39. 17
  40. 5 and 9

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