← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 309 · 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 93²? (a) 7569 (b) 8109 (c) 8649 (d) 8659
  2. 2.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 478 (d) 170
  3. 3.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 256 (d) 16
  4. 4.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 10609 (d) 9991
  5. 5.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
  6. 6.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 169 (d) 25
  7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
  8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 10 (d) 30
  9. 9.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 85 (c) 103 (d) 157
  10. 10.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 25 (d) 81
  11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3584 (d) 3136
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  13. 13.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 100 (d) 142
  14. 14.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 14 (d) 8
  15. 15.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4624 (d) 4904
  16. 16.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 9 (c) 19 (d) 20
  17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
  18. 18.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 49 (c) 1 (d) 25
  19. 19.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  20. 20.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
  21. 21.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 7225 (d) 6375
  22. 22.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 9 (d) 185
  23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 376 (c) 256 (d) 196
  24. 24.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  26. 26.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  27. 27.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 169 (d) 89
  28. 28.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  29. 29.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 7 (c) 65 (d) 49
  30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  31. 31.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
  32. 32.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  33. 33.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  34. 34.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 11 (d) 21
  35. 35.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 901 (c) 841 (d) 971
  36. 36.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 276 (c) 116 (d) 196
  37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9975 (d) 9025
  38. 38.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 36 (d) 4
  39. 39.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 657 (d) 225
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17

Answer key

  1. 8649
  2. 170
  3. 16
  4. 9409
  5. (x + 2)(x + 9)
  6. 25
  7. 30
  8. 11
  9. 85
  10. 25
  11. 3136
  12. (x + 8)(x + 9)
  13. 58
  14. 15
  15. 5184
  16. 20
  17. 14
  18. 1
  19. (x + 3)(x + 9)
  20. 11
  21. 5625
  22. 9
  23. 136
  24. (x + 4)(x + 5)
  25. (x + 5)(x + 7)
  26. (x + 3)(x + 6)
  27. 89
  28. (x + 3)(x + 4)
  29. 49
  30. (x + 3)(x + 5)
  31. 16
  32. 42
  33. (x + 4)(x + 7)
  34. 10
  35. 961
  36. 116
  37. 9025
  38. 4
  39. 225
  40. 72

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