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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 42 · 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 + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  3. 3.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 45 (d) 15
  4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
  5. 5.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  6. 6.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
  7. 7.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 1089 (d) 891
  8. 8.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)
  9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
  11. 11.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  12. 12.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)
  13. 13.If a + b = 6 and ab = 5, what is (a − b)²? (a) 4 (b) 16 (c) 26 (d) 36
  14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 7406 (d) 8836
  16. 16.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  17. 17.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 41 (c) 81 (d) 1
  18. 18.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 13 (c) 32 (d) 11
  19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  20. 20.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 196 (c) 376 (d) 136
  21. 21.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 22 (c) 20 (d) 10
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  23. 23.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 8 (d) 64
  24. 24.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 7056 (c) 5776 (d) 6384
  25. 25.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 7 (d) 10
  26. 26.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 61 (c) 91 (d) 1
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  29. 29.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 88 (d) 40
  30. 30.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 5625 (d) 4225
  31. 31.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2704 (d) 2504
  32. 32.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
  33. 33.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 324 (d) 170
  34. 34.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 429 (d) 149
  35. 35.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 2 (d) 10
  36. 36.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
  37. 37.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1531 (d) 1681
  38. 38.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 361 (d) 9
  39. 39.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 121 (c) 181 (d) 61
  40. 40.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 198 (d) 90

Answer key

  1. (x + 2)(x + 5)
  2. (x + 2)(x + 6)
  3. 14
  4. 56
  5. 1681
  6. 4
  7. 729
  8. (x + 7)(x + 9)
  9. (x + 5)(x + 6)
  10. 4761
  11. (x + 3)(x + 7)
  12. (x + 3)(x + 6)
  13. 16
  14. (x + 5)(x + 9)
  15. 7396
  16. 12
  17. 1
  18. 12
  19. 15
  20. 136
  21. 21
  22. (x + 6)(x + 7)
  23. 64
  24. 5776
  25. 7
  26. 1
  27. 17
  28. (x + 4)(x + 5)
  29. 40
  30. 5625
  31. 2704
  32. 48
  33. 170
  34. 149
  35. 4
  36. 85
  37. 1521
  38. 9
  39. 61
  40. 90

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