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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 924 · 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 are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 2 and 3 (d) 1 and 6
  2. 2.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 10 (d) 100
  3. 3.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 91 (d) 61
  4. 4.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 1 and 10 (d) 3 and 6
  5. 5.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 10 (d) 11
  6. 6.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  7. 7.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 6375 (d) 7225
  8. 8.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 1 and 54 (d) 2 and 27
  9. 9.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 6 and 8 (d) 1 and 13
  10. 10.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1849 (d) 1369
  11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  12. 12.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
  15. 15.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
  16. 16.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
  17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 7579 (d) 8091
  19. 19.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 16
  20. 20.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 180 (d) 468
  21. 21.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  22. 22.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 13 (d) 14
  23. 23.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  24. 24.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 225 (c) 169 (d) 337
  25. 25.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8084 (c) 8836 (d) 7396
  26. 26.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 81 (c) 45 (d) 3
  27. 27.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 9 (c) 8 (d) 15
  28. 28.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 144 (d) 104
  29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 8 (d) 7
  30. 30.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 49 (d) 7
  31. 31.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 313 (d) 181
  32. 32.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 196 (d) 100
  33. 33.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1596 (d) 1764
  34. 34.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 91 (c) 82 (d) 100
  35. 35.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
  36. 36.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2504 (d) 2714
  37. 37.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  38. 38.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 117
  39. 39.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
  40. 40.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 160 (d) 64

Answer key

  1. 2 and 3
  2. 100
  3. 1
  4. 2 and 9
  5. 10
  6. 13
  7. 5625
  8. 6 and 9
  9. 6 and 8
  10. 1849
  11. 17
  12. 7 and 9
  13. (x + 8)(x + 9)
  14. 1
  15. 6 and 7
  16. 16
  17. (x + 4)(x + 5)
  18. 7569
  19. 2 and 8
  20. 180
  21. (x + 5)(x + 7)
  22. 13
  23. (x + 5)(x + 9)
  24. 113
  25. 7396
  26. 9
  27. 8
  28. 104
  29. 8
  30. 49
  31. 137
  32. 4
  33. 1444
  34. 82
  35. 3 and 7
  36. 2704
  37. (x + 4)(x + 8)
  38. 9
  39. 1156
  40. 64

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