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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 793 · 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 sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  2. 2.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  3. 3.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 10 (c) 7 (d) 9
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  5. 5.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 8 (d) 1 and 5
  6. 6.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 144 (c) 104 (d) 64
  7. 7.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  8. 8.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 169 (d) 229
  9. 9.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 6 (c) 1 and 4 (d) 2 and 4
  10. 10.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 67 (d) 53
  11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 7225 (d) 9035
  13. 13.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
  14. 14.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 8 (d) 6
  15. 15.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 36 (d) 20
  16. 16.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 5194 (c) 4624 (d) 4904
  17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 30 (d) 12
  18. 18.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 6384 (c) 7056 (d) 5786
  19. 19.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 266 (d) 384
  20. 20.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
  21. 21.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 2 and 10 (d) 5 and 7
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  23. 23.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 24 (d) 25
  24. 24.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  26. 26.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
  27. 27.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
  28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  29. 29.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 2 (d) 4
  30. 30.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 30 (d) 11
  31. 31.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 7 (d) 9
  32. 32.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 133 (d) 97
  33. 33.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 81 (c) 9 (d) 153
  34. 34.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 451 (d) 361
  35. 35.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 5041 (d) 4901
  36. 36.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1444 (d) 1774
  37. 37.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 7066 (d) 6416
  38. 38.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 9984 (d) 10816
  39. 39.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4771 (c) 4899 (d) 4761

Answer key

  1. 17
  2. 193
  3. 10
  4. 72
  5. 2 and 4
  6. 64
  7. (x + 3)(x + 7)
  8. 169
  9. 2 and 3
  10. 53
  11. (x + 5)(x + 8)
  12. 9025
  13. 181
  14. 7
  15. 20
  16. 5184
  17. 11
  18. 5776
  19. 256
  20. 1
  21. 5 and 7
  22. (x + 6)(x + 7)
  23. 24
  24. (x + 3)(x + 9)
  25. (x + 5)(x + 7)
  26. 164
  27. 130
  28. (x + 7)(x + 9)
  29. 4
  30. 30
  31. 8
  32. 97
  33. 81
  34. 441
  35. 5041
  36. 1764
  37. 7056
  38. 9216
  39. (x + 8)(x + 9)
  40. 4761

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