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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 632 · 40 questions
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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 96²? (a) 9226 (b) 9984 (c) 9216 (d) 10816
  2. 2.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3491 (c) 3599 (d) 3721
  3. 3.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
  4. 4.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10609 (d) 10009
  5. 5.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3979 (d) 3609
  6. 6.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5625 (c) 5635 (d) 7225
  7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  8. 8.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 43 (d) 41
  9. 9.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 106 (c) 151 (d) 196
  10. 10.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2025 (d) 2035
  11. 11.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  12. 12.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 4 (d) 16
  13. 13.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9035 (c) 8125 (d) 9025
  14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  15. 15.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 122 (c) 144 (d) 166
  16. 16.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
  17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
  18. 18.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 8 (d) 12
  19. 19.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 2 and 21 (d) 1 and 42
  20. 20.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 65 (c) 81 (d) 49
  21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
  22. 22.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7225 (d) 7235
  23. 23.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 12 (d) 33
  24. 24.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8
  25. 25.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  26. 26.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 15 (d) 14
  27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
  28. 28.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 9 (c) 10 (d) 21
  29. 29.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 676 (d) 916
  30. 30.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 169 (c) 89 (d) 249
  31. 31.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  32. 32.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 7 (d) 10
  33. 33.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
  34. 34.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
  35. 35.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 10414 (d) 9604
  36. 36.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 34 (d) 49
  37. 37.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
  38. 38.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6409 (c) 5929 (d) 6899
  40. 40.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 10 (c) 22 (d) 16

Answer key

  1. 9216
  2. 3481
  3. 36
  4. 10609
  5. 3969
  6. 5625
  7. 17
  8. 42
  9. 106
  10. 2025
  11. (x + 3)(x + 9)
  12. 16
  13. 9025
  14. (x + 7)(x + 9)
  15. 122
  16. 5 and 6
  17. 14
  18. 7
  19. 6 and 7
  20. 49
  21. 15
  22. 7225
  23. 32
  24. 2 and 4
  25. 208
  26. 15
  27. 181
  28. 10
  29. 1156
  30. 89
  31. (x + 6)(x + 9)
  32. 10
  33. 56
  34. 28
  35. 10404
  36. 34
  37. 5 and 7
  38. (x + 3)(x + 8)
  39. 6889
  40. 10

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