← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 282 · 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 are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 54 (c) 6 and 9 (d) 1 and 14
  2. 2.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 12 (d) 7
  3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  4. 4.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)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5486 (d) 5476
  6. 6.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1691 (c) 1601 (d) 1521
  7. 7.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 10 (d) 11
  8. 8.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
  9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  10. 10.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 121 (d) 65
  11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 5625 (c) 4235 (d) 4875
  14. 14.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  16. 16.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 122 (c) 133 (d) 144
  17. 17.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 539 (d) 409
  18. 18.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 196 (c) 8 (d) 130
  19. 19.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 225 (c) 625 (d) 375
  20. 20.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  21. 21.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 64 (c) 8 (d) 82
  22. 22.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 2035 (c) 1625 (d) 1225
  23. 23.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 113 (d) 337
  24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
  25. 25.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  26. 26.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
  27. 27.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  28. 28.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 429 (d) 219
  29. 29.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 2 (c) 4 (d) 34
  30. 30.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
  31. 31.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  32. 32.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 40 (c) 64 (d) 52
  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.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
  35. 35.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 9 (c) 121 (d) 101
  36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  37. 37.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 4 (d) 324
  38. 38.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 6416 (d) 5776
  40. 40.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 49 (d) 37

Answer key

  1. 6 and 9
  2. 8
  3. 17
  4. (x + 4)(x + 7)
  5. 5476
  6. 1681
  7. 10
  8. 1296
  9. (x + 6)(x + 8)
  10. 65
  11. (x + 6)(x + 7)
  12. (x + 7)(x + 9)
  13. 4225
  14. (x + 5)(x + 8)
  15. (x + 8)(x + 9)
  16. 122
  17. 529
  18. 64
  19. 225
  20. 15
  21. 64
  22. 2025
  23. 113
  24. (x + 2)(x + 3)
  25. (x + 4)(x + 8)
  26. 11
  27. (x + 2)(x + 4)
  28. 149
  29. 4
  30. 16
  31. (x + 7)(x + 8)
  32. 40
  33. (x + 3)(x + 7)
  34. 8 and 9
  35. 81
  36. (x + 2)(x + 9)
  37. 16
  38. (x + 2)(x + 8)
  39. 7056
  40. 25

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