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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 171 · 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 sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  2. 2.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 21 (d) 2 and 8
  3. 3.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1596 (d) 1454
  4. 4.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 100 (d) 76
  5. 5.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 2 and 5 (d) 1 and 5
  6. 6.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 30 (d) 11
  7. 7.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  8. 8.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10826 (d) 10816
  9. 9.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3136 (d) 3584
  10. 10.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
  11. 11.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 3025 (d) 3625
  12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  13. 13.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 16 (d) 4
  14. 14.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 9 (d) 10
  15. 15.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  16. 16.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
  17. 17.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
  18. 18.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 90 (c) 144 (d) 6
  19. 19.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1521 (c) 1691 (d) 1601
  20. 20.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 3025 (c) 2475 (d) 2035
  21. 21.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  22. 22.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 7 (c) 49 (d) 109
  23. 23.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 142 (d) 58
  24. 24.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 169 (c) 5 (d) 97
  25. 25.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
  26. 26.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 5 (c) 7 (d) 1
  27. 27.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 271 (d) 181
  28. 28.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
  29. 29.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 106 (c) 286 (d) 196
  30. 30.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 9 (c) 24 (d) 11
  31. 31.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 13 (d) 9
  32. 32.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 121 (c) 177 (d) 93
  33. 33.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
  34. 34.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 2035 (d) 1225
  35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  37. 37.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
  38. 38.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 14 (d) 15
  39. 39.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)

Answer key

  1. 17
  2. 3 and 7
  3. 1444
  4. 52
  5. 2 and 4
  6. 30
  7. 6
  8. 10816
  9. 3136
  10. 4
  11. 4225
  12. 72
  13. 16
  14. 10
  15. 34
  16. 2809
  17. 7 and 8
  18. 36
  19. 1681
  20. 2025
  21. 8 and 9
  22. 49
  23. 58
  24. 25
  25. 146
  26. 1
  27. 1
  28. 361
  29. 106
  30. 10
  31. 14
  32. 65
  33. 54
  34. 2025
  35. (x + 4)(x + 8)
  36. (x + 2)(x + 7)
  37. 42
  38. 15
  39. (x + 6)(x + 7)
  40. (x + 2)(x + 8)

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