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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 648 · 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 is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 31 (d) 29
  2. 2.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
  3. 3.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  4. 4.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 1 and 13 (d) 5 and 9
  5. 5.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 13 (c) 37 (d) 36
  6. 6.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 169 (c) 73 (d) 121
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 289 (c) 299 (d) 529
  9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 208 (d) 592
  10. 10.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
  11. 11.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 4 (d) 7
  12. 12.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
  14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  15. 15.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 25 (d) 23
  16. 16.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 6391 (d) 5929
  17. 17.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 12 (d) 33
  18. 18.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 9 (d) 225
  19. 19.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
  20. 20.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 13 (d) 32
  21. 21.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 4916 (d) 5476
  22. 22.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4899 (c) 4761 (d) 4771
  23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
  24. 24.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 676 (d) 884
  25. 25.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
  26. 26.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3491 (d) 3721
  27. 27.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 81 (c) 5 (d) 25
  28. 28.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 1 (d) 127
  29. 29.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 5 and 8 (c) 2 and 20 (d) 1 and 12
  30. 30.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 7225 (c) 5625 (d) 5635
  31. 31.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6401 (d) 6561
  32. 32.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  33. 33.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 20 (d) 21
  34. 34.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  35. 35.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 64 (c) 88 (d) 52
  36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 15 (d) 17
  37. 37.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 441
  39. 39.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
  40. 40.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 166 (c) 133 (d) 144

Answer key

  1. 30
  2. 7 and 9
  3. (x + 5)(x + 7)
  4. 5 and 9
  5. 36
  6. 73
  7. (x + 7)(x + 9)
  8. 289
  9. 208
  10. 5 and 7
  11. 5
  12. (x + 6)(x + 8)
  13. 9409
  14. (x + 3)(x + 7)
  15. 24
  16. 5929
  17. 32
  18. 81
  19. 225
  20. 12
  21. 5476
  22. 4761
  23. 17
  24. 676
  25. 2809
  26. 3481
  27. 25
  28. 1
  29. 5 and 8
  30. 5625
  31. 6561
  32. (x + 4)(x + 9)
  33. 21
  34. (x + 5)(x + 8)
  35. 40
  36. 16
  37. (x + 4)(x + 8)
  38. 9
  39. 16
  40. 122

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