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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 547 · 40 questions
TalentJR

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 = 56, what is a² + b²? (a) 225 (b) 337 (c) 169 (d) 113
  2. 2.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 3 and 7 (d) 1 and 9
  3. 3.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 256 (c) 4 (d) 136
  4. 4.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 100 (d) 58
  5. 5.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 7744 (d) 8464
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  8. 8.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
  9. 9.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 1 and 13 (d) 6 and 8
  10. 10.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
  11. 11.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 1166 (d) 916
  12. 12.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  13. 13.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 117 (d) 225
  14. 14.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
  16. 16.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 90 (d) 198
  17. 17.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 91 (d) 61
  18. 18.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 130 (d) 4
  19. 19.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 396 (d) 334
  20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
  21. 21.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 4 and 9 (c) 1 and 36 (d) 1 and 12
  22. 22.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 63 (d) 81
  23. 23.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  24. 24.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)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3259 (d) 3969
  26. 26.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
  27. 27.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 145 (c) 169 (d) 121
  28. 28.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 125 (d) 9
  29. 29.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9025 (d) 9975
  30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  31. 31.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  32. 32.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  34. 34.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 88 (d) 40
  35. 35.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 15 (d) 7
  36. 36.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 409 (d) 539
  37. 37.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 3 and 9 (c) 2 and 10 (d) 1 and 11
  38. 38.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
  39. 39.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  40. 40.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 85 (c) 169 (d) 127

Answer key

  1. 113
  2. 3 and 7
  3. 16
  4. 58
  5. 8464
  6. (x + 2)(x + 4)
  7. (x + 3)(x + 4)
  8. 4
  9. 6 and 8
  10. 8649
  11. 1156
  12. 1
  13. 9
  14. 1
  15. 72
  16. 90
  17. 1
  18. 4
  19. 324
  20. 17
  21. 4 and 9
  22. 45
  23. 146
  24. (x + 5)(x + 8)
  25. 3249
  26. 4
  27. 121
  28. 81
  29. 9025
  30. (x + 8)(x + 9)
  31. (x + 6)(x + 8)
  32. (x + 4)(x + 5)
  33. (x + 7)(x + 8)
  34. 40
  35. 8
  36. 529
  37. 3 and 9
  38. 148
  39. (x + 5)(x + 7)
  40. 85

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