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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 160 · 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.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
  2. 2.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 34 (d) 4
  3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  4. 4.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)
  5. 5.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 18 (d) 11
  6. 6.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 65 (c) 93 (d) 121
  7. 7.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  8. 8.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 253 (c) 169 (d) 127
  9. 9.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 2 and 4 (d) 1 and 4
  10. 10.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
  11. 11.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 45 (c) 14 (d) 44
  12. 12.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10826 (d) 10016
  13. 13.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 146 (c) 256 (d) 366
  14. 14.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3616 (d) 3136
  15. 15.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
  16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
  17. 17.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 130 (d) 2
  18. 18.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
  19. 19.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
  20. 20.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
  21. 21.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 1 and 10 (c) 2 and 5 (d) 3 and 4
  22. 22.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)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 7225 (d) 6375
  24. 24.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 10 (d) 11
  25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3249 (d) 3259
  27. 27.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 12 (c) 14 (d) 13
  28. 28.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  29. 29.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 9 (c) 149 (d) 3
  30. 30.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 169 (d) 157
  31. 31.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  32. 32.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
  33. 33.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 7225 (c) 9035 (d) 8125
  34. 34.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
  35. 35.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 6 (c) 144 (d) 90
  36. 36.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6561 (d) 6401
  37. 37.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 17 (d) 3
  38. 38.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 1089 (d) 729
  39. 39.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 1 and 56 (d) 2 and 28
  40. 40.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)

Answer key

  1. 6889
  2. 4
  3. (x + 2)(x + 3)
  4. (x + 7)(x + 8)
  5. 11
  6. 65
  7. (x + 5)(x + 9)
  8. 85
  9. 2 and 3
  10. 10
  11. 45
  12. 10816
  13. 146
  14. 4096
  15. 42
  16. 15
  17. 4
  18. 9
  19. 113
  20. 4356
  21. 2 and 5
  22. (x + 4)(x + 7)
  23. 5625
  24. 11
  25. (x + 3)(x + 9)
  26. 3249
  27. 13
  28. 14
  29. 9
  30. 145
  31. (x + 3)(x + 8)
  32. 5 and 6
  33. 9025
  34. 16
  35. 36
  36. 6561
  37. 9
  38. 729
  39. 7 and 8
  40. (x + 4)(x + 9)

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