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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 727 · 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 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  3. 3.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3136 (d) 3584
  6. 6.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 7 (d) 9
  7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 196 (d) 156
  8. 8.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
  9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3249 (d) 3609
  12. 12.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 529 (d) 361
  13. 13.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  14. 14.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 10 (d) 30
  15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 598 (d) 400
  16. 16.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 16 (c) 8 (d) 15
  17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 729 (c) 891 (d) 739
  19. 19.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 235 (c) 625 (d) 375
  20. 20.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744
  21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6409 (c) 5929 (d) 6899
  23. 23.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 91 (d) 61
  24. 24.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  25. 25.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  26. 26.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
  27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  28. 28.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 36 (c) 116 (d) 196
  29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
  30. 30.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 909 (d) 1099
  31. 31.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  32. 32.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9984 (c) 9216 (d) 9226
  33. 33.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 43 (d) 37
  34. 34.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 1 (d) 25
  35. 35.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 4235 (d) 3025
  36. 36.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
  37. 37.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
  39. 39.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
  40. 40.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 361 (d) 441

Answer key

  1. 1681
  2. 17
  3. (x + 2)(x + 5)
  4. (x + 7)(x + 8)
  5. 3136
  6. 8
  7. 116
  8. 4
  9. (x + 2)(x + 6)
  10. 576
  11. 3969
  12. 193
  13. (x + 2)(x + 4)
  14. 11
  15. 202
  16. 15
  17. (x + 8)(x + 9)
  18. 729
  19. 225
  20. 8464
  21. 13
  22. 6889
  23. 61
  24. 113
  25. 56
  26. 1
  27. (x + 5)(x + 8)
  28. 36
  29. 130
  30. 1089
  31. 42
  32. 9216
  33. 37
  34. 1
  35. 4225
  36. (x + 3)(x + 9)
  37. (x + 5)(x + 6)
  38. 1296
  39. 73
  40. 361

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