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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 3 · 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 − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
  2. 2.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 5184 (d) 4904
  3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1296 (c) 1616 (d) 1946
  4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 170 (d) 16
  5. 5.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)
  6. 6.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 1 and 18 (c) 2 and 9 (d) 3 and 6
  7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
  8. 8.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 289 (d) 217
  9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
  10. 10.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  11. 11.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  12. 12.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
  13. 13.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
  14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
  15. 15.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 31 (c) 30 (d) 29
  16. 16.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 7931 (d) 8099
  17. 17.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 53 (c) 81 (d) 109
  18. 18.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
  19. 19.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  20. 20.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
  21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  22. 22.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 11 (d) 10
  23. 23.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
  24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  25. 25.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6724 (d) 6084
  26. 26.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 3 (c) 1 and 6 (d) 2 and 4
  27. 27.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 1 (d) 49
  28. 28.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 116 (d) 156
  29. 29.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 1 and 10 (d) 3 and 8
  30. 30.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 101 (c) 121 (d) 111
  31. 31.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
  32. 32.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 61 (c) 91 (d) 181
  33. 33.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 13 (d) 11
  34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
  35. 35.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  36. 36.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  37. 37.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 7 (d) 1
  38. 38.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4761 (c) 4771 (d) 5041
  39. 39.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 4 (d) 20
  40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)

Answer key

  1. 16
  2. 5184
  3. 1936
  4. 16
  5. (x + 5)(x + 8)
  6. 3 and 6
  7. (x + 3)(x + 5)
  8. 1
  9. 185
  10. (x + 5)(x + 9)
  11. 56
  12. 5476
  13. 36
  14. 17
  15. 30
  16. 7921
  17. 53
  18. 9
  19. 8836
  20. 74
  21. (x + 3)(x + 8)
  22. 10
  23. 961
  24. (x + 8)(x + 9)
  25. 6724
  26. 2 and 3
  27. 1
  28. 116
  29. 3 and 8
  30. 101
  31. 225
  32. 61
  33. 12
  34. 6084
  35. (x + 2)(x + 4)
  36. (x + 4)(x + 6)
  37. 1
  38. 4761
  39. 4
  40. (x + 5)(x + 6)

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