← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 664 · 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 = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 81 (d) 61
  2. 2.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 169 (c) 213 (d) 125
  3. 3.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 180 (d) 252
  4. 4.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2475 (d) 2025
  5. 5.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 10
  6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  7. 7.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 85 (d) 121
  8. 8.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13
  9. 9.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 19 (d) 37
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
  11. 11.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)
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  13. 13.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 121 (d) 145
  14. 14.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 313 (c) 225 (d) 137
  15. 15.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 247 (d) 170
  16. 16.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
  17. 17.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 121 (c) 169 (d) 97
  18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  19. 19.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 63 (c) 81 (d) 117
  20. 20.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 130 (c) 196 (d) 262
  21. 21.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 5 (d) 7
  22. 22.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 1 and 13 (d) 3 and 15
  23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  24. 24.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5329 (c) 4909 (d) 5339
  25. 25.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 1 and 7 (d) 3 and 5
  26. 26.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 676 (d) 1166
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 4356 (d) 5476
  30. 30.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  31. 31.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 43 (d) 37
  32. 32.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3259 (c) 3591 (d) 3969
  33. 33.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
  34. 34.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 169 (d) 129
  35. 35.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
  36. 36.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1235 (d) 1225
  37. 37.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
  38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
  39. 39.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 20 (d) 9
  40. 40.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 69 (d) 29

Answer key

  1. 41
  2. 125
  3. 180
  4. 2025
  5. 24
  6. 17
  7. 49
  8. 6 and 8
  9. 13
  10. 16
  11. (x + 3)(x + 5)
  12. (x + 6)(x + 7)
  13. 121
  14. 137
  15. 170
  16. 841
  17. 73
  18. (x + 4)(x + 7)
  19. 45
  20. 130
  21. 1
  22. 5 and 9
  23. (x + 5)(x + 6)
  24. 5329
  25. 3 and 5
  26. (x + 4)(x + 6)
  27. 1156
  28. (x + 8)(x + 9)
  29. 4356
  30. (x + 2)(x + 7)
  31. 37
  32. 3249
  33. 42
  34. 89
  35. 5625
  36. 1225
  37. 10
  38. (x + 5)(x + 9)
  39. 20
  40. 29

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