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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 32 · 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) 17 (c) 16 (d) 15
  2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 4 (d) 2
  3. 3.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 7 (d) 2 and 6
  4. 4.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3364 (c) 3854 (d) 3604
  5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  6. 6.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 164 (d) 324
  7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 3025 (c) 3625 (d) 4235
  9. 9.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 56 (d) 15
  11. 11.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
  12. 12.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
  13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  14. 14.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 7 (d) 6
  15. 15.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  16. 16.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  17. 17.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
  18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  19. 19.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 31 (d) 33
  20. 20.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193
  21. 21.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6241 (d) 6251
  22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 598 (d) 202
  24. 24.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 85 (c) 169 (d) 127
  25. 25.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 189 (d) 153
  26. 26.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2025 (d) 2475
  27. 27.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 40 (c) 13 (d) 41
  28. 28.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 16 (d) 170
  29. 29.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5339 (c) 5329 (d) 4909
  30. 30.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
  31. 31.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
  32. 32.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  34. 34.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
  35. 35.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 9 (d) 7
  36. 36.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 19 (d) 13
  37. 37.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 15 (d) 14
  38. 38.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
  39. 39.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6889 (c) 5929 (d) 6899
  40. 40.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 61 (d) 81

Answer key

  1. 16
  2. 4
  3. 2 and 6
  4. 3844
  5. (x + 7)(x + 8)
  6. 164
  7. (x + 2)(x + 3)
  8. 4225
  9. 1681
  10. 15
  11. 1521
  12. 4
  13. (x + 5)(x + 7)
  14. 8
  15. 113
  16. 361
  17. 676
  18. 17
  19. 32
  20. 25
  21. 6241
  22. (x + 6)(x + 9)
  23. 202
  24. 1
  25. 153
  26. 2025
  27. 40
  28. 16
  29. 5329
  30. 9
  31. 56
  32. (x + 3)(x + 5)
  33. (x + 8)(x + 9)
  34. (x + 3)(x + 9)
  35. 8
  36. 13
  37. 15
  38. 1936
  39. 6889
  40. 41

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