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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 69 · 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 = 7 and ab = 10, what is a² + b²? (a) 39 (b) 29 (c) 49 (d) 69
  2. 2.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  3. 3.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 1 and 15 (d) 3 and 21
  4. 4.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13
  5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  6. 6.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 12 (c) 5 and 8 (d) 1 and 40
  7. 7.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
  8. 8.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 15 (d) 54
  9. 9.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 15 (d) 13
  10. 10.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6404 (d) 6734
  11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  12. 12.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 21
  13. 13.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)
  14. 14.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
  15. 15.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 43 (d) 13
  16. 16.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
  17. 17.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  19. 19.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 169 (c) 139 (d) 109
  20. 20.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
  21. 21.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 2 (d) 4
  22. 22.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 9 (d) 8
  23. 23.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 10
  24. 24.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
  25. 25.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)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2304 (d) 2704
  27. 27.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
  28. 28.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  29. 29.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3481 (d) 3601
  30. 30.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 5476 (d) 4366
  31. 31.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  32. 32.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 64 (d) 50
  33. 33.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 3 (c) 89 (d) 9
  34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  35. 35.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 376 (c) 196 (d) 256
  36. 36.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 157 (c) 169 (d) 193
  37. 37.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 10 (d) 11
  38. 38.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 12 (d) 13
  39. 39.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
  40. 40.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5

Answer key

  1. 29
  2. (x + 5)(x + 6)
  3. 7 and 9
  4. 1
  5. (x + 7)(x + 8)
  6. 5 and 8
  7. (x + 2)(x + 5)
  8. 54
  9. 14
  10. 6724
  11. 9
  12. 3 and 7
  13. (x + 8)(x + 9)
  14. 8 and 9
  15. 42
  16. 74
  17. 1
  18. 15
  19. 109
  20. 185
  21. 4
  22. 9
  23. 2 and 9
  24. 4
  25. (x + 5)(x + 7)
  26. 2704
  27. 9
  28. 10
  29. 3721
  30. 4356
  31. 4225
  32. 50
  33. 9
  34. (x + 3)(x + 8)
  35. 136
  36. 145
  37. 11
  38. 13
  39. 961
  40. 3 and 5

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