← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 529 · 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.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 144 (d) 2
  2. 2.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
  3. 3.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
  4. 4.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 139 (c) 229 (d) 169
  5. 5.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  6. 6.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1859 (d) 1609
  7. 7.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
  8. 8.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 399 (d) 441
  9. 9.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  10. 10.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 34 (d) 49
  11. 11.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 10 (d) 8
  12. 12.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
  15. 15.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 65 (d) 9
  16. 16.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5339 (d) 5329
  17. 17.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 3 (d) 361
  18. 18.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10609 (d) 10009
  19. 19.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6571 (d) 6241
  20. 20.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1764 (d) 1604
  21. 21.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 109 (c) 81 (d) 53
  22. 22.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
  23. 23.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4225 (c) 5625 (d) 4925
  24. 24.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 8 (d) 7
  25. 25.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 17 (c) 19 (d) 18
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  28. 28.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 1156 (c) 884 (d) 686
  29. 29.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 13 (c) 9 (d) 14
  30. 30.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 625 (d) 425
  31. 31.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 5 and 7 (d) 1 and 35
  32. 32.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  33. 33.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 13 (d) 32
  34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  35. 35.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3364 (d) 3374
  36. 36.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 7056 (d) 6384
  37. 37.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8659 (d) 8649
  38. 38.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 9
  39. 39.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 901 (d) 971
  40. 40.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3599 (c) 3491 (d) 3481

Answer key

  1. 4
  2. 42
  3. 7 and 8
  4. 109
  5. 146
  6. 1849
  7. 63
  8. 361
  9. 9025
  10. 34
  11. 9
  12. (x + 3)(x + 5)
  13. (x + 7)(x + 8)
  14. (x + 2)(x + 6)
  15. 9
  16. 5329
  17. 9
  18. 10609
  19. 6561
  20. 1764
  21. 53
  22. 529
  23. 5625
  24. 7
  25. 18
  26. (x + 8)(x + 9)
  27. 17
  28. 676
  29. 14
  30. 625
  31. 5 and 7
  32. (x + 2)(x + 4)
  33. 12
  34. 72
  35. 3364
  36. 5776
  37. 8649
  38. 10
  39. 961
  40. 3481

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