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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 914 · 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 − 12x + 32 = 0? (a) 33 (b) 12 (c) 31 (d) 32
  2. 2.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 4 and 9 (c) 2 and 18 (d) 1 and 36
  3. 3.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 28 (c) 11 (d) 27
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  5. 5.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  6. 6.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 85 (d) 121
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 256 (d) 193
  8. 8.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 169 (d) 129
  9. 9.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8659 (d) 8649
  10. 10.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
  11. 11.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 81 (c) 121 (d) 61
  12. 12.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2496 (d) 2304
  13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  14. 14.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 9
  15. 15.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
  16. 16.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5476 (d) 5486
  17. 17.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 57 (d) 55
  18. 18.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6399 (d) 6241
  19. 19.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  20. 20.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 4096 (d) 3584
  21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
  22. 22.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
  23. 23.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  24. 24.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 64 (d) 50
  25. 25.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 3 and 8 (c) 1 and 24 (d) 1 and 10
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  28. 28.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
  29. 29.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 54 (d) 15
  30. 30.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 10 (d) 196
  31. 31.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1764 (d) 1454
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  33. 33.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
  34. 34.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 971 (d) 901
  35. 35.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 53 (c) 109 (d) 67
  36. 36.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10619 (d) 10009
  37. 37.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 61 (c) 91 (d) 181
  38. 38.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 76 (d) 100
  39. 39.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 49 (d) 25
  40. 40.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 9025 (d) 7225

Answer key

  1. 32
  2. 4 and 9
  3. 28
  4. (x + 8)(x + 9)
  5. (x + 2)(x + 3)
  6. 85
  7. 130
  8. 89
  9. 8649
  10. 5625
  11. 41
  12. 2304
  13. (x + 7)(x + 8)
  14. 8
  15. 3844
  16. 5476
  17. 56
  18. 6241
  19. 6
  20. 3136
  21. 225
  22. (x + 2)(x + 8)
  23. (x + 4)(x + 9)
  24. 36
  25. 3 and 8
  26. (x + 2)(x + 7)
  27. 72
  28. 1
  29. 54
  30. 100
  31. 1444
  32. (x + 6)(x + 9)
  33. 8 and 9
  34. 961
  35. 53
  36. 10609
  37. 61
  38. 52
  39. 25
  40. 9025

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