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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 725 · 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 + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  2. 2.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
  3. 3.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 49 (d) 7
  4. 4.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 14 (d) 48
  5. 5.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 784 (d) 1034
  6. 6.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)
  7. 7.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 4 and 9 (c) 1 and 36 (d) 2 and 18
  8. 8.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
  9. 9.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 37 (d) 35
  10. 10.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 28 (c) 1 and 10 (d) 4 and 7
  11. 11.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 361 (d) 9
  12. 12.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  13. 13.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 30 (d) 12
  14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  15. 15.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  16. 16.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 64 (d) 8
  17. 17.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2916 (d) 2116
  18. 18.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
  19. 19.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 4761 (d) 5041
  20. 20.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
  22. 22.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 875 (d) 625
  23. 23.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
  24. 24.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 7406 (d) 8836
  26. 26.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 625 (d) 925
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  28. 28.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 352 (d) 256
  29. 29.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  30. 30.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 145 (d) 217
  32. 32.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  33. 33.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 36 (c) 116 (d) 6
  34. 34.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 97 (d) 241
  35. 35.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 37 (c) 43 (d) 49
  36. 36.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 10 (c) 17 (d) 16
  37. 37.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 3 and 6 (d) 2 and 7
  38. 38.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 63 (d) 45
  39. 39.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 14 (d) 13
  40. 40.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 256 (c) 6 (d) 146

Answer key

  1. (x + 7)(x + 8)
  2. 13
  3. 49
  4. 14
  5. 1024
  6. (x + 7)(x + 9)
  7. 4 and 9
  8. 10201
  9. 36
  10. 4 and 7
  11. 9
  12. (x + 4)(x + 6)
  13. 11
  14. (x + 5)(x + 9)
  15. 15
  16. 64
  17. 2116
  18. 130
  19. 5041
  20. 7 and 8
  21. 41
  22. 625
  23. 3721
  24. (x + 6)(x + 8)
  25. 7396
  26. 1225
  27. 17
  28. 160
  29. 16
  30. 8 and 9
  31. 145
  32. 1089
  33. 36
  34. 97
  35. 37
  36. 16
  37. 2 and 7
  38. 45
  39. 14
  40. 36

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