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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 698 · 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.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
  2. 2.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
  3. 3.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)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 5939 (c) 6391 (d) 6889
  5. 5.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
  6. 6.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  7. 7.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19
  8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  9. 9.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 36 (c) 324 (d) 180
  10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  11. 11.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 5 (c) 53 (d) 25
  12. 12.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 729 (d) 739
  13. 13.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 289 (c) 391 (d) 529
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
  15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 10 (d) 8
  16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  17. 17.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 23 (d) 24
  18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 1 (d) 181
  19. 19.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 193 (d) 529
  20. 20.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 31 (c) 36 (d) 46
  21. 21.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 25 (d) 49
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  23. 23.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  24. 24.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 15 (d) 9
  25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  27. 27.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 19 (c) 1 (d) 13
  28. 28.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 8091 (d) 7579
  29. 29.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 64 (d) 8
  30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
  32. 32.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)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
  34. 34.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 4 and 6 (c) 2 and 12 (d) 1 and 9
  35. 35.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 7056 (d) 5786
  36. 36.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
  37. 37.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9226 (c) 9984 (d) 10816
  38. 38.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 375 (d) 225
  39. 39.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 1 and 56 (d) 2 and 28
  40. 40.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10609 (d) 10619

Answer key

  1. (x + 3)(x + 4)
  2. 164
  3. (x + 3)(x + 6)
  4. 5929
  5. 6 and 7
  6. (x + 2)(x + 6)
  7. 18
  8. 72
  9. 36
  10. (x + 5)(x + 8)
  11. 25
  12. 729
  13. 289
  14. 208
  15. 9
  16. 225
  17. 24
  18. 1
  19. 193
  20. 26
  21. 1
  22. (x + 7)(x + 9)
  23. 52
  24. 14
  25. (x + 2)(x + 8)
  26. 181
  27. 1
  28. 7569
  29. 64
  30. (x + 6)(x + 8)
  31. 1681
  32. (x + 5)(x + 9)
  33. 5041
  34. 4 and 6
  35. 5776
  36. 9
  37. 9216
  38. 225
  39. 7 and 8
  40. 10609

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