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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 560 · 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.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5635 (d) 5625
  3. 3.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)
  4. 4.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 61 (d) 91
  5. 5.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 10 (d) 8
  6. 6.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
  7. 7.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
  8. 8.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 15 (d) 48
  9. 9.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 12 (d) 13
  10. 10.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
  11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  12. 12.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 88 (d) 52
  13. 13.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 100 (c) 118 (d) 82
  14. 14.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 12
  15. 15.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
  16. 16.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3969 (d) 3249
  17. 17.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 21 (d) 1 and 9
  18. 18.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 125 (d) 175
  19. 19.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 25 (d) 13
  20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  21. 21.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 27 (c) 1 and 11 (d) 3 and 9
  22. 22.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 100 (d) 52
  23. 23.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
  24. 24.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 10 (d) 1 and 18
  25. 25.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
  26. 26.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 85 (d) 103
  27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  28. 28.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
  29. 29.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 36 (c) 26 (d) 16
  30. 30.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
  31. 31.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6396 (d) 6094
  32. 32.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 784 (d) 1034
  33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  34. 34.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
  35. 35.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 130 (d) 4
  36. 36.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 151 (d) 106
  37. 37.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 10 (d) 11
  38. 38.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
  39. 39.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  40. 40.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 6 (c) 196 (d) 36

Answer key

  1. (x + 2)(x + 3)
  2. 5625
  3. (x + 7)(x + 8)
  4. 61
  5. 7
  6. 1
  7. 961
  8. 14
  9. 13
  10. 72
  11. (x + 7)(x + 9)
  12. 40
  13. 82
  14. 11
  15. 2025
  16. 3969
  17. 3 and 7
  18. 125
  19. 13
  20. 181
  21. 3 and 9
  22. 52
  23. 1
  24. 2 and 9
  25. 484
  26. 85
  27. (x + 4)(x + 5)
  28. 116
  29. 16
  30. 9
  31. 6084
  32. 1024
  33. (x + 6)(x + 7)
  34. 37
  35. 4
  36. 106
  37. 11
  38. 6889
  39. (x + 2)(x + 4)
  40. 36

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