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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 243 · 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 + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  3. 3.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 13 (d) 14
  4. 4.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
  5. 5.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 6
  6. 6.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
  7. 7.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 11 (d) 145
  8. 8.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 13 (d) 27
  9. 9.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8836 (d) 8846
  11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 11025 (c) 9975 (d) 9035
  13. 13.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 1 and 8 (c) 2 and 7 (d) 3 and 6
  14. 14.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
  15. 15.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  16. 16.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 1156 (d) 916
  17. 17.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2496 (d) 2304
  18. 18.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 9 (c) 17 (d) 18
  19. 19.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3969 (d) 3591
  20. 20.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 4761 (d) 5041
  21. 21.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  23. 23.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 63 (d) 1 and 15
  24. 24.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 904 (d) 1024
  25. 25.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 625 (d) 635
  26. 26.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 121 (d) 61
  27. 27.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  29. 29.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
  30. 30.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
  31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1379 (d) 1369
  33. 33.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2504 (d) 2304
  34. 34.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  35. 35.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
  36. 36.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
  37. 37.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 576 (d) 256
  38. 38.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 5929 (d) 6889
  39. 39.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 117 (c) 144 (d) 198
  40. 40.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 11 (c) 10 (d) 24

Answer key

  1. (x + 6)(x + 9)
  2. (x + 7)(x + 8)
  3. 13
  4. 42
  5. 7
  6. 73
  7. 121
  8. 12
  9. (x + 2)(x + 9)
  10. 8836
  11. (x + 2)(x + 4)
  12. 9025
  13. 2 and 7
  14. 9
  15. 484
  16. 1156
  17. 2304
  18. 18
  19. 3249
  20. 5041
  21. (x + 3)(x + 9)
  22. 17
  23. 7 and 9
  24. 1024
  25. 625
  26. 41
  27. 56
  28. (x + 8)(x + 9)
  29. 72
  30. 324
  31. 1
  32. 1369
  33. 2704
  34. (x + 2)(x + 3)
  35. 6 and 7
  36. 7225
  37. 576
  38. 6889
  39. 90
  40. 11

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