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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 318 · 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.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 9025 (d) 11025
  2. 2.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 7 (c) 65 (d) 81
  3. 3.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
  4. 4.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
  5. 5.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)
  6. 6.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 10 (c) 7 (d) 11
  7. 7.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 5635 (d) 4225
  8. 8.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 9 (d) 1 and 16
  9. 9.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
  10. 10.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
  11. 11.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 45 (d) 63
  12. 12.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 146 (d) 256
  13. 13.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 8 (d) 7
  14. 14.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 29 (c) 49 (d) 39
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  16. 16.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 14 (d) 15
  17. 17.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
  18. 18.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)
  19. 19.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  20. 20.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  21. 21.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 10816 (d) 9226
  22. 22.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1601 (c) 1681 (d) 1521
  23. 23.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  24. 24.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 225 (d) 117
  25. 25.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3259 (d) 3969
  26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2704 (c) 2304 (d) 2714
  27. 27.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 109 (c) 214 (d) 144
  28. 28.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
  29. 29.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  30. 30.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  31. 31.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 4 and 5 (c) 1 and 8 (d) 2 and 10
  32. 32.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 13 (c) 14 (d) 15
  33. 33.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 25 (c) 3 (d) 17
  34. 34.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 539 (d) 409
  35. 35.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 401 (d) 441
  36. 36.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 130 (d) 256
  37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  38. 38.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 121 (d) 181
  39. 39.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 10
  40. 40.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1609 (c) 1369 (d) 1859

Answer key

  1. 11025
  2. 49
  3. 125
  4. 137
  5. (x + 7)(x + 8)
  6. 10
  7. 5625
  8. 2 and 8
  9. 81
  10. 7396
  11. 45
  12. 146
  13. 7
  14. 29
  15. 17
  16. 15
  17. 9
  18. (x + 3)(x + 7)
  19. 4
  20. 15
  21. 9216
  22. 1681
  23. (x + 3)(x + 5)
  24. 117
  25. 3249
  26. 2704
  27. 74
  28. (x + 3)(x + 6)
  29. (x + 4)(x + 8)
  30. 36
  31. 4 and 5
  32. 14
  33. 9
  34. 529
  35. 441
  36. 130
  37. (x + 4)(x + 5)
  38. 61
  39. 24
  40. 1849

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