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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 554 · 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 = 16 and ab = 55, what is a² + b²? (a) 366 (b) 146 (c) 201 (d) 256
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
  3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  4. 4.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 17 (c) 3 (d) 25
  5. 5.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 43 (c) 37 (d) 61
  6. 6.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 91 (d) 100
  7. 7.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 63 (d) 15
  9. 9.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 7 (d) 8
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  11. 11.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
  12. 12.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8101 (d) 8291
  13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  14. 14.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 63 (d) 45
  15. 15.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10001 (c) 9801 (d) 10201
  16. 16.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 41 (d) 42
  17. 17.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 33 (c) 21 (d) 25
  18. 18.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
  19. 19.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 4225 (d) 3625
  20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 891 (d) 729
  22. 22.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  23. 23.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 225 (c) 333 (d) 171
  24. 24.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4925 (c) 5635 (d) 4225
  25. 25.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 7 (d) 12
  27. 27.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 6375 (d) 7225
  28. 28.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  29. 29.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 48 (d) 47
  30. 30.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 7 (d) 5
  31. 31.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 241 (c) 169 (d) 133
  32. 32.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 6 (d) 2 and 6
  33. 33.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  34. 34.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 256 (d) 130
  35. 35.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 109 (d) 74
  36. 36.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 64 (c) 8 (d) 104
  37. 37.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
  38. 38.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 657 (d) 441
  39. 39.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 416 (d) 586
  40. 40.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)

Answer key

  1. 146
  2. 17
  3. (x + 2)(x + 3)
  4. 9
  5. 37
  6. 82
  7. (x + 4)(x + 9)
  8. 16
  9. 6
  10. (x + 8)(x + 9)
  11. 145
  12. 8281
  13. 56
  14. 45
  15. 10201
  16. 42
  17. 17
  18. 16
  19. 4225
  20. (x + 4)(x + 5)
  21. 729
  22. (x + 2)(x + 6)
  23. 117
  24. 5625
  25. (x + 5)(x + 6)
  26. 7
  27. 5625
  28. (x + 3)(x + 8)
  29. 48
  30. 5
  31. 97
  32. 3 and 4
  33. (x + 6)(x + 8)
  34. 4
  35. 74
  36. 64
  37. 841
  38. 225
  39. 576
  40. (x + 3)(x + 9)

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