← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 777 · 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.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1599 (d) 1681
  2. 2.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4225 (c) 4235 (d) 5625
  3. 3.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 160 (c) 64 (d) 256
  4. 4.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 5 (d) 289
  5. 5.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 109 (c) 49 (d) 169
  6. 6.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
  7. 7.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 6 (c) 1 and 4 (d) 2 and 3
  8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
  9. 9.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
  10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
  11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  12. 12.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  13. 13.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
  14. 14.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 4356 (d) 5476
  15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
  16. 16.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
  17. 17.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 891 (d) 1089
  18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  19. 19.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 116 (d) 196
  20. 20.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)
  21. 21.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  22. 22.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 4225 (d) 3625
  23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4891 (d) 4489
  26. 26.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  27. 27.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 15 (c) 9 (d) 14
  28. 28.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 8836 (d) 7396
  29. 29.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
  30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  31. 31.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 148 (d) 100
  32. 32.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 324 (d) 484
  33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  34. 34.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 16 (d) 62
  35. 35.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1575 (d) 1225
  36. 36.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 12 (d) 10
  37. 37.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 9409 (d) 10009
  39. 39.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 181 (d) 121
  40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)

Answer key

  1. 1521
  2. 4225
  3. 64
  4. 25
  5. 49
  6. 7 and 8
  7. 2 and 3
  8. 72
  9. 45
  10. 1089
  11. 17
  12. 8 and 9
  13. 5476
  14. 4356
  15. (x + 2)(x + 9)
  16. 145
  17. 729
  18. (x + 7)(x + 8)
  19. 116
  20. (x + 8)(x + 9)
  21. 2 and 8
  22. 4225
  23. (x + 6)(x + 8)
  24. (x + 5)(x + 6)
  25. 4489
  26. (x + 4)(x + 8)
  27. 14
  28. 7396
  29. 56
  30. (x + 5)(x + 9)
  31. 100
  32. 484
  33. 15
  34. 63
  35. 1225
  36. 11
  37. (x + 3)(x + 6)
  38. 10609
  39. 61
  40. (x + 5)(x + 8)

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