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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 462 · 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 are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
  2. 2.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 5929 (d) 6391
  3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1296 (d) 1946
  4. 4.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
  5. 5.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
  6. 6.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
  7. 7.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
  8. 8.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 256 (d) 266
  10. 10.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
  11. 11.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
  12. 12.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 1 (d) 25
  13. 13.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 8 (c) 9 (d) 10
  14. 14.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  15. 15.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 97 (d) 73
  16. 16.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 1 (d) 49
  17. 17.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 12 (d) 13
  18. 18.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)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6241 (d) 6401
  20. 20.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 9 (d) 361
  21. 21.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8846 (c) 8116 (d) 8836
  22. 22.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4761 (c) 5051 (d) 4901
  23. 23.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 324 (d) 396
  24. 24.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 11
  25. 25.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 441 (d) 399
  26. 26.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 31 (c) 36 (d) 26
  27. 27.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
  28. 28.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 2 and 10 (c) 1 and 27 (d) 1 and 11
  29. 29.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 49 (c) 169 (d) 109
  30. 30.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
  31. 31.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 297 (d) 153
  32. 32.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
  33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
  35. 35.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 3025 (c) 3625 (d) 4235
  36. 36.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  38. 38.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
  39. 39.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 15 (d) 17
  40. 40.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9811 (c) 9801 (d) 10201

Answer key

  1. 6 and 7
  2. 5929
  3. 1936
  4. 7225
  5. 784
  6. (x + 5)(x + 8)
  7. 7
  8. (x + 5)(x + 9)
  9. 256
  10. 170
  11. 15
  12. 1
  13. 9
  14. 181
  15. 73
  16. 1
  17. 12
  18. (x + 3)(x + 6)
  19. 6561
  20. 9
  21. 8836
  22. 5041
  23. 324
  24. 24
  25. 361
  26. 26
  27. 10404
  28. 3 and 9
  29. 49
  30. 961
  31. 153
  32. 4
  33. (x + 7)(x + 9)
  34. 3025
  35. 4225
  36. 225
  37. (x + 2)(x + 7)
  38. 113
  39. 16
  40. 9801

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