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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 505 · 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 + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  2. 2.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
  3. 3.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 166 (c) 144 (d) 122
  4. 4.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  5. 5.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 12 (d) 26
  6. 6.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)
  7. 7.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 11 (c) 13 (d) 8
  8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
  9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 273 (d) 537
  10. 10.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 9 (d) 81
  11. 11.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
  12. 12.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
  13. 13.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2219 (c) 2491 (d) 2809
  14. 14.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2611 (d) 2601
  15. 15.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 6375 (c) 5625 (d) 5635
  16. 16.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8836 (c) 8846 (d) 8116
  17. 17.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8659 (d) 8649
  19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  20. 20.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 196 (d) 100
  21. 21.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 235 (c) 625 (d) 375
  23. 23.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 11 (c) 24 (d) 10
  24. 24.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 286 (d) 196
  25. 25.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 784 (d) 794
  26. 26.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 7056 (d) 5776
  27. 27.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 7744 (d) 8096
  28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  29. 29.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
  31. 31.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
  32. 32.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 52 (d) 20
  33. 33.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 891 (d) 729
  34. 34.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
  35. 35.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
  36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  37. 37.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 13 (d) 36
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
  39. 39.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 149 (c) 3 (d) 289
  40. 40.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12

Answer key

  1. (x + 8)(x + 9)
  2. 16
  3. 122
  4. (x + 3)(x + 9)
  5. 27
  6. (x + 7)(x + 8)
  7. 12
  8. 8281
  9. 185
  10. 9
  11. 1936
  12. 2704
  13. 2209
  14. 2601
  15. 5625
  16. 8836
  17. (x + 2)(x + 9)
  18. 8649
  19. (x + 4)(x + 8)
  20. 4
  21. (x + 2)(x + 7)
  22. 225
  23. 11
  24. 106
  25. 784
  26. 5776
  27. 7744
  28. (x + 5)(x + 6)
  29. 56
  30. 17
  31. 6 and 7
  32. 20
  33. 729
  34. 82
  35. (x + 3)(x + 7)
  36. 72
  37. 13
  38. 11
  39. 9
  40. 12

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