← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 405 · 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 = 15 and ab = 36, what is a² + b²? (a) 153 (b) 297 (c) 225 (d) 189
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  3. 3.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 904 (c) 1024 (d) 784
  5. 5.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)
  6. 6.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 40 (c) 1 and 12 (d) 2 and 20
  7. 7.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 9 (d) 24
  8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
  9. 9.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  10. 10.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61
  11. 11.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 84 (d) 68
  12. 12.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 14 (d) 16
  13. 13.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 11025 (c) 10025 (d) 9025
  14. 14.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 784 (d) 794
  15. 15.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  16. 16.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3035 (c) 3025 (d) 4225
  17. 17.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
  18. 18.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 157 (c) 103 (d) 85
  19. 19.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127
  20. 20.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
  21. 21.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  22. 22.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
  23. 23.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 686 (d) 884
  24. 24.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 156 (d) 196
  25. 25.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  27. 27.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 6425 (d) 7225
  30. 30.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
  31. 31.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 41 (c) 13 (d) 39
  32. 32.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3136 (c) 4096 (d) 3616
  33. 33.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5635 (c) 4925 (d) 5625
  34. 34.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  35. 35.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 13 (c) 5 (d) 9
  36. 36.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8101 (d) 8291
  37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  38. 38.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 169 (c) 3 (d) 89
  39. 39.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 45 (d) 81
  40. 40.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)

Answer key

  1. 153
  2. (x + 2)(x + 6)
  3. (x + 2)(x + 7)
  4. 1024
  5. (x + 8)(x + 9)
  6. 5 and 8
  7. 10
  8. 1
  9. 149
  10. 61
  11. 68
  12. 15
  13. 11025
  14. 784
  15. 16
  16. 3025
  17. 1089
  18. 85
  19. 85
  20. 16
  21. (x + 5)(x + 7)
  22. 6 and 9
  23. 676
  24. 116
  25. 4
  26. 181
  27. (x + 2)(x + 8)
  28. (x + 6)(x + 7)
  29. 7225
  30. 13
  31. 40
  32. 4096
  33. 5625
  34. 15
  35. 5
  36. 8281
  37. (x + 6)(x + 9)
  38. 9
  39. 45
  40. (x + 3)(x + 7)

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