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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 232 · 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.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 225 (d) 333
  2. 2.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 7225 (d) 5625
  3. 3.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  4. 4.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 25 (d) 24
  5. 5.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
  6. 6.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
  7. 7.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521
  10. 10.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 8 (d) 5
  11. 11.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 20 (c) 28 (d) 52
  12. 12.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 4 (d) 20
  13. 13.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
  14. 14.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 2 and 3 (d) 1 and 4
  15. 15.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
  16. 16.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4624 (d) 4896
  17. 17.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 8 (c) 7 (d) 12
  18. 18.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
  19. 19.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 10 (c) 100 (d) 144
  20. 20.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
  21. 21.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9999 (d) 9801
  22. 22.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 53 (d) 25
  23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
  26. 26.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 5 (c) 13 (d) 9
  27. 27.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  28. 28.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 6 and 9 (d) 2 and 27
  29. 29.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
  30. 30.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 81 (d) 41
  31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  32. 32.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 113 (d) 225
  33. 33.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 961 (d) 841
  34. 34.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  35. 35.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
  36. 36.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 17 (d) 18
  37. 37.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 4 (d) 2
  38. 38.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 116 (d) 36
  39. 39.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 34 (d) 36
  40. 40.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561

Answer key

  1. 117
  2. 7225
  3. (x + 2)(x + 7)
  4. 24
  5. 56
  6. 8 and 9
  7. (x + 3)(x + 9)
  8. (x + 6)(x + 7)
  9. 1521
  10. 6
  11. 20
  12. 4
  13. 2304
  14. 2 and 3
  15. 1764
  16. 4624
  17. 7
  18. 3 and 6
  19. 100
  20. 5625
  21. 9801
  22. 25
  23. (x + 8)(x + 9)
  24. (x + 2)(x + 5)
  25. 72
  26. 5
  27. (x + 4)(x + 7)
  28. 6 and 9
  29. 1444
  30. 41
  31. (x + 5)(x + 6)
  32. 113
  33. 961
  34. 576
  35. 9
  36. 18
  37. 4
  38. 36
  39. 35
  40. 6241

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