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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 23 · 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.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  2. 2.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 3 and 7 (c) 2 and 8 (d) 1 and 9
  3. 3.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)
  4. 4.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217
  5. 5.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13
  6. 6.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 5625 (d) 7235
  7. 7.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3025 (d) 3625
  8. 8.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
  9. 9.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
  10. 10.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
  11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 101 (c) 121 (d) 111
  13. 13.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 9 (d) 185
  14. 14.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
  15. 15.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 5329 (d) 4891
  16. 16.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 13 (d) 37
  17. 17.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
  18. 18.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 7 (d) 8
  19. 19.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 244 (d) 164
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  21. 21.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 20 (c) 36 (d) 28
  22. 22.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
  23. 23.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 299 (d) 289
  24. 24.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 384 (d) 256
  25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  26. 26.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 26 (c) 4 (d) 36
  27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  29. 29.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 37 (d) 49
  30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  31. 31.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 177 (d) 93
  32. 32.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 1 and 24 (d) 2 and 12
  33. 33.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)
  34. 34.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 8 (c) 196 (d) 64
  35. 35.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2704 (d) 2304
  36. 36.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 5184 (d) 4904
  37. 37.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904
  38. 38.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 441 (d) 361
  39. 39.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 4 (d) 208
  40. 40.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 5 (d) 7

Answer key

  1. (x + 4)(x + 7)
  2. 2 and 8
  3. (x + 5)(x + 6)
  4. 145
  5. 6 and 8
  6. 7225
  7. 4225
  8. 42
  9. 6889
  10. (x + 3)(x + 7)
  11. (x + 6)(x + 9)
  12. 101
  13. 9
  14. 8 and 9
  15. 4489
  16. 13
  17. 63
  18. 7
  19. 164
  20. (x + 5)(x + 7)
  21. 20
  22. 12
  23. 289
  24. 256
  25. (x + 3)(x + 9)
  26. 16
  27. (x + 4)(x + 9)
  28. 72
  29. 25
  30. (x + 8)(x + 9)
  31. 65
  32. 3 and 8
  33. (x + 7)(x + 8)
  34. 64
  35. 2704
  36. 5184
  37. 1024
  38. 441
  39. 16
  40. 1

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