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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 263 · 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 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8291 (d) 8101
  3. 3.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
  4. 4.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 3136 (d) 4096
  5. 5.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 5329 (c) 4891 (d) 4499
  6. 6.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 13 (c) 9 (d) 7
  7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  9. 9.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 10 (d) 14
  10. 10.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
  11. 11.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1584 (d) 1936
  13. 13.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 361 (d) 181
  14. 14.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 153 (c) 189 (d) 297
  15. 15.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
  16. 16.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
  17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
  18. 18.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 961 (d) 841
  19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  20. 20.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 289 (c) 25 (d) 157
  21. 21.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 4 (c) 16 (d) 324
  22. 22.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 875 (d) 625
  23. 23.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 3 (d) 9
  24. 24.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2916 (d) 2116
  25. 25.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
  26. 26.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)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1936 (d) 1616
  28. 28.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 4 and 6 (d) 1 and 9
  29. 29.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 17 (d) 21
  30. 30.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
  31. 31.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 3 and 6 (c) 2 and 7 (d) 1 and 8
  32. 32.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
  33. 33.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 116 (d) 36
  34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 8096 (d) 7744
  35. 35.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 6 (d) 5
  36. 36.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  37. 37.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 7 (c) 49 (d) 121
  38. 38.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 34 (c) 4 (d) 64
  39. 39.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
  40. 40.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 37 (c) 49 (d) 61

Answer key

  1. 1521
  2. 8281
  3. 2 and 8
  4. 4096
  5. 4489
  6. 5
  7. (x + 5)(x + 8)
  8. (x + 3)(x + 4)
  9. 9
  10. 6561
  11. 8 and 9
  12. 1296
  13. 181
  14. 153
  15. (x + 3)(x + 8)
  16. 11
  17. 11
  18. 961
  19. 72
  20. 25
  21. 16
  22. 625
  23. 9
  24. 2116
  25. 1444
  26. (x + 7)(x + 8)
  27. 1936
  28. 4 and 6
  29. 17
  30. 1681
  31. 2 and 7
  32. 145
  33. 36
  34. 7744
  35. 6
  36. 6 and 7
  37. 49
  38. 4
  39. 21
  40. 37

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