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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 961 · 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.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8649 (d) 8091
  3. 3.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 74 (d) 214
  4. 4.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 2 and 28 (d) 1 and 14
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
  6. 6.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3599 (d) 3481
  7. 7.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)
  8. 8.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 106 (d) 151
  9. 9.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 13 (d) 41
  10. 10.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 100 (d) 2
  11. 11.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  12. 12.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 9 (c) 17 (d) 18
  13. 13.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  14. 14.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 113 (d) 225
  15. 15.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3364 (d) 3844
  16. 16.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 9 (c) 14 (d) 13
  17. 17.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 52 (d) 2
  18. 18.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 2 and 5 (d) 1 and 6
  19. 19.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
  20. 20.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 5625 (c) 4875 (d) 4225
  21. 21.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 73 (d) 65
  22. 22.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 26 (d) 46
  23. 23.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
  24. 24.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1369 (c) 1591 (d) 1849
  25. 25.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
  26. 26.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3136 (c) 3146 (d) 3584
  27. 27.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 1 and 8 (c) 2 and 4 (d) 2 and 5
  28. 28.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 219 (d) 429
  29. 29.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
  30. 30.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 784 (d) 1024
  31. 31.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 382 (d) 130
  32. 32.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
  33. 33.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
  34. 34.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 6 (d) 4
  35. 35.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6251 (d) 6561
  36. 36.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 1 (d) 41
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  38. 38.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  39. 39.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
  40. 40.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 41 (d) 39

Answer key

  1. (x + 7)(x + 9)
  2. 7569
  3. 74
  4. 7 and 8
  5. 15
  6. 3481
  7. (x + 3)(x + 9)
  8. 106
  9. 42
  10. 4
  11. (x + 3)(x + 7)
  12. 18
  13. 36
  14. 113
  15. 3844
  16. 14
  17. 4
  18. 2 and 5
  19. 1681
  20. 4225
  21. 65
  22. 26
  23. 25
  24. 1369
  25. 484
  26. 3136
  27. 2 and 4
  28. 149
  29. 136
  30. 1024
  31. 130
  32. (x + 2)(x + 5)
  33. 16
  34. 5
  35. 6241
  36. 1
  37. (x + 4)(x + 7)
  38. 4
  39. 1936
  40. 40

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