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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 155 · 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.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 13 (d) 43
  2. 2.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 8 (d) 7
  3. 3.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 6 and 9 (d) 2 and 27
  4. 4.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  5. 5.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  6. 6.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 36 (d) 196
  7. 7.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 4235 (d) 3025
  8. 8.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 16 (d) 256
  9. 9.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 7569 (d) 8091
  10. 10.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 4225 (c) 3575 (d) 3025
  11. 11.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
  12. 12.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 68 (c) 6 (d) 36
  13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  14. 14.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 73 (d) 97
  15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  16. 16.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 6 (d) 8
  17. 17.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 35 (c) 37 (d) 13
  18. 18.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
  19. 19.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  22. 22.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
  23. 23.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 1 and 10 (d) 2 and 15
  24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  25. 25.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
  26. 26.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 10 (d) 1 and 18
  27. 27.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  28. 28.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 229 (d) 109
  29. 29.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
  30. 30.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 40 (d) 12
  31. 31.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 7 (d) 8
  32. 32.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
  33. 33.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
  35. 35.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 7 (d) 49
  36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1444 (d) 1764
  38. 38.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 196 (d) 4
  39. 39.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 166 (c) 133 (d) 144
  40. 40.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3491 (d) 3721

Answer key

  1. 42
  2. 6
  3. 6 and 9
  4. (x + 3)(x + 9)
  5. (x + 5)(x + 6)
  6. 36
  7. 4225
  8. 16
  9. 7569
  10. 3025
  11. 3025
  12. 36
  13. (x + 6)(x + 8)
  14. 73
  15. 1
  16. 7
  17. 36
  18. 4
  19. (x + 5)(x + 8)
  20. (x + 5)(x + 7)
  21. 225
  22. 8649
  23. 5 and 6
  24. (x + 7)(x + 9)
  25. 185
  26. 2 and 9
  27. (x + 6)(x + 7)
  28. 109
  29. (x + 2)(x + 4)
  30. 13
  31. 8
  32. 4 and 8
  33. 63
  34. 56
  35. 49
  36. (x + 8)(x + 9)
  37. 1764
  38. 16
  39. 122
  40. 3481

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