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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 451 · 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 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096
  2. 2.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  3. 3.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 49 (c) 69 (d) 29
  4. 4.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 4 and 5 (d) 1 and 20
  5. 5.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 63 (d) 45
  6. 6.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 478 (d) 170
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  8. 8.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)
  9. 9.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
  10. 10.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  11. 11.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)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
  13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4624 (d) 4634
  16. 16.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 169 (c) 133 (d) 241
  17. 17.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 244 (d) 196
  18. 18.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
  19. 19.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 1 and 35 (d) 2 and 10
  20. 20.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1681 (d) 1599
  21. 21.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9035 (d) 9025
  23. 23.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  24. 24.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  25. 25.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 73 (d) 97
  26. 26.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)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 5625 (c) 4875 (d) 4225
  28. 28.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 91 (d) 181
  29. 29.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9614 (d) 9996
  31. 31.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 153 (c) 297 (d) 225
  32. 32.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
  33. 33.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9984 (c) 9216 (d) 10816
  34. 34.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 41 (d) 42
  35. 35.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 109 (c) 81 (d) 67
  36. 36.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6
  37. 37.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
  38. 38.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 451 (d) 361
  39. 39.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 37 (d) 35
  40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10404 (c) 9604 (d) 10004

Answer key

  1. 4096
  2. (x + 4)(x + 7)
  3. 29
  4. 4 and 5
  5. 45
  6. 170
  7. (x + 7)(x + 9)
  8. (x + 3)(x + 9)
  9. 6 and 7
  10. (x + 2)(x + 7)
  11. (x + 8)(x + 9)
  12. 1296
  13. (x + 6)(x + 8)
  14. (x + 6)(x + 7)
  15. 4624
  16. 97
  17. 148
  18. 4
  19. 5 and 7
  20. 1521
  21. (x + 5)(x + 6)
  22. 9025
  23. 6
  24. (x + 4)(x + 8)
  25. 73
  26. (x + 2)(x + 8)
  27. 4225
  28. 61
  29. (x + 6)(x + 9)
  30. 9604
  31. 153
  32. 16
  33. 9216
  34. 42
  35. 53
  36. 3 and 5
  37. 12
  38. 441
  39. 36
  40. 10404

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