← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 609 · 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.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 13 (d) 45
  2. 2.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  3. 3.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 17 (c) 11 (d) 19
  4. 4.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 313 (d) 137
  5. 5.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 9999 (d) 10201
  6. 6.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 7569 (d) 8659
  7. 7.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 91 (d) 100
  8. 8.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
  9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  10. 10.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 8 (d) 6
  11. 11.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 8 (c) 82 (d) 100
  12. 12.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
  13. 13.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  14. 14.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 49 (d) 64
  15. 15.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 5041 (d) 4761
  17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 361 (d) 441
  18. 18.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8649 (c) 7569 (d) 8091
  20. 20.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 55 (d) 57
  21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 1024 (d) 784
  22. 22.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 3 (c) 29 (d) 49
  23. 23.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5339 (c) 4489 (d) 5329
  24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 81 (c) 61 (d) 41
  25. 25.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 1089 (d) 891
  26. 26.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 31 (c) 36 (d) 46
  27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  28. 28.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 122 (c) 133 (d) 144
  29. 29.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 40 (c) 64 (d) 52
  30. 30.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 9025 (d) 11025
  31. 31.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 6 (d) 1 and 12
  32. 32.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
  33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  35. 35.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)
  36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 6 (d) 5
  37. 37.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1166 (d) 1156
  38. 38.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
  39. 39.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 10 (d) 9
  40. 40.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 404 (d) 494

Answer key

  1. 14
  2. (x + 3)(x + 8)
  3. 18
  4. 137
  5. 9801
  6. 8649
  7. 82
  8. 841
  9. 1089
  10. 8
  11. 64
  12. 9
  13. (x + 3)(x + 9)
  14. 34
  15. (x + 4)(x + 5)
  16. 4761
  17. 441
  18. (x + 3)(x + 6)
  19. 7569
  20. 56
  21. 1024
  22. 9
  23. 5329
  24. 41
  25. 729
  26. 26
  27. (x + 7)(x + 9)
  28. 122
  29. 40
  30. 9025
  31. 3 and 4
  32. (x + 2)(x + 9)
  33. (x + 3)(x + 7)
  34. (x + 5)(x + 6)
  35. (x + 7)(x + 8)
  36. 5
  37. 1156
  38. 15
  39. 10
  40. 484

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