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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 601 · 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 are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 1 and 6 (d) 3 and 4
  2. 2.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  3. 3.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 112 (c) 144 (d) 208
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  5. 5.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 253 (d) 85
  6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 193 (d) 382
  8. 8.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
  9. 9.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
  10. 10.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 9 (d) 14
  11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  12. 12.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 5 (c) 125 (d) 25
  13. 13.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 121 (c) 65 (d) 93
  14. 14.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 45 (d) 63
  15. 15.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8474 (c) 8104 (d) 7744
  17. 17.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 13 (c) 14 (d) 9
  18. 18.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
  19. 19.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
  20. 20.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
  21. 21.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
  22. 22.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 9 (d) 10
  23. 23.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6241 (d) 6561
  24. 24.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 88 (d) 64
  25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  27. 27.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 10 (d) 3 and 6
  28. 28.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5184 (c) 5194 (d) 4904
  29. 29.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
  30. 30.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 196 (d) 106
  31. 31.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
  32. 32.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
  34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  35. 35.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 76 (d) 100
  36. 36.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  37. 37.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
  38. 38.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 244 (d) 484
  39. 39.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
  40. 40.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2601 (d) 2499

Answer key

  1. 3 and 4
  2. (x + 5)(x + 8)
  3. 80
  4. 17
  5. 85
  6. (x + 5)(x + 7)
  7. 130
  8. 63
  9. 54
  10. 9
  11. (x + 6)(x + 7)
  12. 25
  13. 65
  14. 45
  15. (x + 2)(x + 6)
  16. 8464
  17. 14
  18. 48
  19. 1369
  20. 89
  21. 961
  22. 10
  23. 6561
  24. 40
  25. (x + 2)(x + 7)
  26. 6
  27. 2 and 9
  28. 5184
  29. 6 and 7
  30. 106
  31. (x + 4)(x + 6)
  32. (x + 3)(x + 5)
  33. 5929
  34. (x + 3)(x + 8)
  35. 52
  36. 529
  37. 1
  38. 164
  39. 8649
  40. 2401

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