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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 985 · 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 is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 45 (d) 44
  2. 2.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 6 (d) 8
  3. 3.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
  4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  5. 5.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)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7056 (d) 7066
  7. 7.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 32 (d) 33
  8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 10 (d) 11
  9. 9.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  10. 10.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 8099 (d) 7931
  12. 12.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1691 (c) 1601 (d) 1521
  13. 13.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 65 (c) 3 (d) 9
  14. 14.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 9025 (c) 8075 (d) 7235
  15. 15.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 164 (d) 484
  16. 16.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 121 (c) 141 (d) 111
  17. 17.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  18. 18.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 9226 (d) 10816
  19. 19.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  20. 20.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 9 (d) 8
  21. 21.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 40 (d) 52
  22. 22.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 100 (d) 91
  23. 23.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 45 (d) 9
  24. 24.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 12 (d) 13
  25. 25.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 289 (d) 433
  26. 26.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 132 (c) 84 (d) 100
  27. 27.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 100 (c) 68 (d) 6
  28. 28.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 36 (c) 6 (d) 196
  29. 29.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 7 (c) 49 (d) 65
  30. 30.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 11025 (c) 9975 (d) 9035
  31. 31.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)
  32. 32.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  33. 33.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
  34. 34.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 196 (d) 106
  35. 35.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
  36. 36.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
  37. 37.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
  39. 39.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16

Answer key

  1. 45
  2. 7
  3. 64
  4. (x + 2)(x + 6)
  5. (x + 7)(x + 8)
  6. 7056
  7. 32
  8. 11
  9. (x + 2)(x + 4)
  10. (x + 6)(x + 8)
  11. 7921
  12. 1681
  13. 9
  14. 7225
  15. 164
  16. 101
  17. 181
  18. 9216
  19. 1
  20. 8
  21. 40
  22. 82
  23. 9
  24. 12
  25. 145
  26. 68
  27. 36
  28. 36
  29. 49
  30. 9025
  31. (x + 6)(x + 7)
  32. 36
  33. 21
  34. 16
  35. 6 and 8
  36. (x + 2)(x + 3)
  37. 2 and 3
  38. (x + 5)(x + 9)
  39. (x + 3)(x + 8)
  40. 16

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