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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 974 · 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.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
  2. 2.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 39 (d) 41
  3. 3.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 164 (d) 2
  4. 4.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9991 (d) 9409
  5. 5.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 7 (d) 9
  6. 6.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 26 (c) 4 (d) 36
  7. 7.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
  8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  9. 9.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 13 (d) 19
  10. 10.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 16 (d) 14
  11. 11.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2314 (c) 2704 (d) 2496
  12. 12.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 8 (d) 7
  13. 13.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
  14. 14.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  15. 15.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 149 (d) 3
  16. 16.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  17. 17.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 32 (c) 1 and 11 (d) 2 and 16
  18. 18.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8836 (c) 8846 (d) 8116
  19. 19.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 404 (d) 484
  20. 20.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 2 (d) 4
  21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  22. 22.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 28 (c) 12 (d) 27
  23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  24. 24.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
  25. 25.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
  26. 26.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
  27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  28. 28.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  29. 29.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  30. 30.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 169 (d) 229
  31. 31.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 1 and 20 (d) 4 and 5
  32. 32.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 61 (d) 1
  33. 33.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 2 and 7 (c) 1 and 14 (d) 3 and 6
  34. 34.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
  35. 35.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 170 (c) 247 (d) 324
  36. 36.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  37. 37.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 11 (c) 9 (d) 7
  38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
  39. 39.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
  40. 40.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 7 (d) 49

Answer key

  1. 8 and 9
  2. 40
  3. 4
  4. 9409
  5. 1
  6. 16
  7. 4
  8. (x + 7)(x + 9)
  9. 13
  10. 15
  11. 2304
  12. 8
  13. 25
  14. (x + 5)(x + 8)
  15. 9
  16. (x + 6)(x + 7)
  17. 4 and 8
  18. 8836
  19. 484
  20. 4
  21. (x + 7)(x + 8)
  22. 27
  23. (x + 8)(x + 9)
  24. 15
  25. 63
  26. 1
  27. (x + 6)(x + 9)
  28. (x + 6)(x + 8)
  29. (x + 5)(x + 7)
  30. 169
  31. 4 and 5
  32. 1
  33. 2 and 7
  34. 529
  35. 170
  36. 89
  37. 10
  38. 1
  39. 225
  40. 49

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