← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 555 · 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 = 50, what is (a − b)²? (a) 125 (b) 225 (c) 5 (d) 25
  2. 2.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 82 (c) 100 (d) 118
  3. 3.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1575 (d) 1235
  4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  6. 6.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 175 (c) 225 (d) 125
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 193 (d) 256
  8. 8.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 49 (c) 85 (d) 121
  9. 9.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  10. 10.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 177 (d) 121
  11. 11.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40
  12. 12.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  13. 13.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 19 (d) 17
  14. 14.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
  15. 15.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6241 (d) 6561
  16. 16.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 3625 (d) 3025
  17. 17.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 21 (d) 10
  18. 18.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 9 (d) 7
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  20. 20.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 15 (d) 45
  21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  22. 22.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 85 (c) 127 (d) 169
  23. 23.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
  24. 24.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1235 (d) 1225
  25. 25.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 25 (d) 49
  26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  28. 28.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 289
  29. 29.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6561 (d) 6401
  30. 30.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 13 (d) 36
  31. 31.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 40 (d) 13
  32. 32.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  33. 33.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 73 (d) 65
  34. 34.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1681 (d) 1521
  35. 35.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
  36. 36.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 164 (c) 324 (d) 2
  37. 37.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2401 (c) 2601 (d) 2499
  38. 38.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 112 (c) 80 (d) 208
  39. 39.If a + b = 14 and ab = 40, what is (a − b)²? (a) 6 (b) 196 (c) 116 (d) 36
  40. 40.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4875 (d) 4235

Answer key

  1. 25
  2. 82
  3. 1225
  4. (x + 2)(x + 6)
  5. 17
  6. 125
  7. 130
  8. 49
  9. (x + 3)(x + 5)
  10. 65
  11. 40
  12. (x + 5)(x + 8)
  13. 18
  14. 181
  15. 6241
  16. 4225
  17. 21
  18. 8
  19. (x + 8)(x + 9)
  20. 14
  21. (x + 7)(x + 8)
  22. 85
  23. 9
  24. 1225
  25. 1
  26. (x + 6)(x + 9)
  27. 10404
  28. 49
  29. 6561
  30. 36
  31. 13
  32. 484
  33. 65
  34. 1681
  35. 9
  36. 4
  37. 2401
  38. 80
  39. 36
  40. 4225

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