← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 313 · 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 73²? (a) 5329 (b) 4909 (c) 5339 (d) 4489
  2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 64 (d) 62
  5. 5.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 391 (c) 299 (d) 529
  6. 6.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 55 (d) 54
  7. 7.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  8. 8.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 441 (d) 657
  9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  10. 10.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 21 (d) 19
  11. 11.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 63 (d) 81
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  13. 13.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
  14. 14.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 153 (c) 189 (d) 297
  15. 15.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 61 (c) 91 (d) 121
  16. 16.If a + b = 13 and ab = 30, what is a² + b²? (a) 109 (b) 139 (c) 229 (d) 169
  17. 17.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1521 (d) 1681
  18. 18.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
  19. 19.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 225 (d) 81
  20. 20.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 8 (d) 7
  21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 81 (c) 121 (d) 41
  22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6899 (d) 6889
  23. 23.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  24. 24.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 184 (c) 104 (d) 144
  25. 25.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
  26. 26.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10001 (d) 10201
  28. 28.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 2 (c) 34 (d) 64
  29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  30. 30.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)
  31. 31.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 1 and 42 (d) 2 and 21
  32. 32.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1936 (d) 1584
  33. 33.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 136 (c) 196 (d) 256
  34. 34.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 17 (c) 9 (d) 3
  35. 35.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 9 (d) 3
  36. 36.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 18 (d) 10
  37. 37.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 18 (d) 12
  38. 38.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 4 (d) 2
  39. 39.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 9 (d) 15
  40. 40.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 40 (d) 52

Answer key

  1. 5329
  2. (x + 2)(x + 4)
  3. 17
  4. 63
  5. 289
  6. 54
  7. 1681
  8. 225
  9. (x + 3)(x + 8)
  10. 20
  11. 45
  12. (x + 2)(x + 8)
  13. 2 and 4
  14. 153
  15. 1
  16. 109
  17. 1521
  18. (x + 4)(x + 6)
  19. 81
  20. 6
  21. 41
  22. 6889
  23. (x + 3)(x + 5)
  24. 104
  25. 3364
  26. (x + 5)(x + 7)
  27. 10201
  28. 4
  29. (x + 7)(x + 8)
  30. (x + 8)(x + 9)
  31. 6 and 7
  32. 1296
  33. 136
  34. 9
  35. 9
  36. 9
  37. 11
  38. 4
  39. 14
  40. 40

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