← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 38 · 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 + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9226 (c) 9216 (d) 10816
  3. 3.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  5. 5.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 6391 (d) 5939
  6. 6.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8836 (c) 8084 (d) 7406
  7. 7.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 117 (d) 81
  8. 8.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 15 (d) 7
  9. 9.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1764 (d) 1444
  11. 11.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 11
  12. 12.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 13 (d) 40
  13. 13.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 91 (d) 61
  14. 14.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
  16. 16.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 44 (d) 45
  17. 17.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 79 (d) 58
  18. 18.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6899 (d) 6889
  19. 19.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2025 (d) 2035
  20. 20.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
  21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 81 (c) 121 (d) 61
  22. 22.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 7921 (d) 8281
  23. 23.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 441 (d) 371
  24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  25. 25.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 1 and 7 (d) 2 and 6
  26. 26.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
  27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  28. 28.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 69 (c) 49 (d) 29
  29. 29.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 25 (c) 21 (d) 17
  30. 30.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2314 (c) 2704 (d) 2304
  31. 31.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 5
  32. 32.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)
  33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 289 (d) 3
  34. 34.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 13 (c) 9 (d) 7
  35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 635 (d) 875
  37. 37.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 90 (d) 144
  38. 38.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2516 (d) 2926
  39. 39.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8474 (c) 7744 (d) 8104
  40. 40.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096

Answer key

  1. (x + 7)(x + 8)
  2. 9216
  3. 5625
  4. 72
  5. 5929
  6. 7396
  7. 45
  8. 8
  9. (x + 5)(x + 7)
  10. 1444
  11. 24
  12. 13
  13. 61
  14. (x + 3)(x + 8)
  15. (x + 4)(x + 7)
  16. 45
  17. 58
  18. 6889
  19. 2025
  20. 2025
  21. 41
  22. 8281
  23. 361
  24. (x + 7)(x + 9)
  25. 2 and 6
  26. 841
  27. (x + 4)(x + 5)
  28. 29
  29. 17
  30. 2304
  31. 25
  32. (x + 4)(x + 8)
  33. 9
  34. 5
  35. (x + 5)(x + 9)
  36. 625
  37. 90
  38. 2916
  39. 8464
  40. 4096

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad