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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 299 · 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.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 15 (c) 16 (d) 17
  2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 794 (c) 1024 (d) 896
  3. 3.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
  4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  6. 6.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 1 (d) 271
  7. 7.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 11 (d) 10
  8. 8.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 7569 (d) 8649
  9. 9.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
  11. 11.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 34 (d) 49
  12. 12.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 196 (c) 130 (d) 64
  13. 13.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
  14. 14.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  15. 15.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 9025 (c) 11035 (d) 10025
  16. 16.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 425 (d) 635
  17. 17.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 376 (c) 136 (d) 256
  18. 18.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
  19. 19.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1034 (c) 904 (d) 1024
  20. 20.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7579 (d) 7569
  21. 21.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 14 (d) 1 and 56
  22. 22.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 169 (c) 127 (d) 85
  23. 23.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2025 (c) 2035 (d) 1625
  24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  25. 25.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 6 and 8 (d) 1 and 13
  26. 26.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 241 (d) 97
  27. 27.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 57 (c) 78 (d) 64
  28. 28.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  29. 29.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 9 (d) 3
  30. 30.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 9025 (d) 8125
  31. 31.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8464 (d) 8096
  32. 32.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 53 (d) 81
  33. 33.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 10 (c) 1 and 24 (d) 2 and 12
  34. 34.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
  35. 35.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3249 (d) 3969
  36. 36.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1296 (d) 1306
  37. 37.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
  38. 38.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4891 (c) 4489 (d) 5329
  39. 39.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 8 (d) 7
  40. 40.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401

Answer key

  1. 16
  2. 784
  3. 10404
  4. (x + 5)(x + 6)
  5. (x + 5)(x + 9)
  6. 1
  7. 11
  8. 8649
  9. 4
  10. 15
  11. 34
  12. 64
  13. 529
  14. 8 and 9
  15. 11025
  16. 625
  17. 136
  18. 9
  19. 1024
  20. 7569
  21. 7 and 8
  22. 1
  23. 2025
  24. (x + 2)(x + 3)
  25. 6 and 8
  26. 97
  27. 50
  28. 56
  29. 9
  30. 9025
  31. 7744
  32. 53
  33. 3 and 8
  34. 225
  35. 3969
  36. 1296
  37. 208
  38. 4489
  39. 7
  40. 6561

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