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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 483 · 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 73²? (a) 5339 (b) 4489 (c) 4909 (d) 5329
  2. 2.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 64 (c) 34 (d) 2
  3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  4. 4.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)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6409 (d) 6899
  6. 6.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  7. 7.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)
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  9. 9.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 9 (d) 17
  10. 10.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
  11. 11.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
  12. 12.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  15. 15.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 13 (d) 32
  16. 16.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  18. 18.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6724 (d) 6734
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  20. 20.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 130 (c) 196 (d) 262
  21. 21.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  22. 22.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 21 (d) 1 and 9
  23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  24. 24.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 286 (d) 106
  25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  26. 26.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 16 (d) 256
  27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  28. 28.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 100 (d) 36
  29. 29.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
  30. 30.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  31. 31.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 2 and 12 (c) 4 and 6 (d) 1 and 24
  32. 32.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 10 (d) 12
  33. 33.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 592 (d) 400
  34. 34.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 49 (d) 64
  35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  36. 36.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 676 (d) 916
  37. 37.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 961 (c) 901 (d) 841
  38. 38.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
  39. 39.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)

Answer key

  1. 5329
  2. 4
  3. 16
  4. (x + 6)(x + 9)
  5. 6889
  6. 4225
  7. (x + 3)(x + 5)
  8. (x + 6)(x + 8)
  9. 9
  10. 7 and 9
  11. 56
  12. 5
  13. (x + 7)(x + 8)
  14. (x + 2)(x + 3)
  15. 12
  16. 529
  17. 17
  18. 6724
  19. (x + 8)(x + 9)
  20. 130
  21. 1024
  22. 3 and 7
  23. (x + 5)(x + 6)
  24. 106
  25. 72
  26. 16
  27. (x + 2)(x + 6)
  28. 36
  29. (x + 2)(x + 9)
  30. (x + 2)(x + 8)
  31. 4 and 6
  32. 11
  33. 208
  34. 34
  35. (x + 7)(x + 9)
  36. 1156
  37. 961
  38. 20
  39. 145
  40. (x + 6)(x + 7)

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