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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 624 · 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 43²? (a) 1849 (b) 1609 (c) 1859 (d) 1369
  2. 2.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10609 (c) 9409 (d) 10619
  3. 3.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 64 (c) 8 (d) 144
  4. 4.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5476 (c) 4356 (d) 5486
  5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  6. 6.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
  7. 7.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  8. 8.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 196
  9. 9.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1235 (d) 1225
  10. 10.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 1156 (d) 676
  11. 11.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 2025 (c) 3035 (d) 3025
  12. 12.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4489 (c) 5329 (d) 4499
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  14. 14.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
  15. 15.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 113 (d) 337
  16. 16.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  17. 17.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
  18. 18.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9801 (d) 9999
  19. 19.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 277 (d) 361
  20. 20.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)
  21. 21.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)
  22. 22.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8
  23. 23.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 169 (d) 97
  24. 24.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  25. 25.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  26. 26.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 9
  27. 27.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 9 (d) 3
  28. 28.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9409 (c) 10609 (d) 9419
  29. 29.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 9801 (d) 10211
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
  31. 31.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 225 (d) 325
  32. 32.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1604 (d) 1764
  33. 33.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 208 (d) 352
  34. 34.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  35. 35.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 24 (d) 11
  37. 37.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9984 (d) 9226
  38. 38.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10404 (d) 10004
  39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 185 (d) 273
  40. 40.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)

Answer key

  1. 1849
  2. 10609
  3. 64
  4. 5476
  5. (x + 6)(x + 8)
  6. 25
  7. 16
  8. 4
  9. 1225
  10. 676
  11. 3025
  12. 4489
  13. 72
  14. 1
  15. 113
  16. (x + 5)(x + 7)
  17. 3 and 7
  18. 9801
  19. 193
  20. (x + 7)(x + 8)
  21. (x + 8)(x + 9)
  22. 2 and 4
  23. 97
  24. (x + 3)(x + 4)
  25. 36
  26. 10
  27. 9
  28. 9409
  29. 10201
  30. 15
  31. 125
  32. 1764
  33. 160
  34. 289
  35. 4225
  36. 11
  37. 9216
  38. 10404
  39. 185
  40. (x + 5)(x + 6)

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