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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 81 · 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.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 16 (d) 4
  2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 13 (d) 15
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  4. 4.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
  6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2496 (c) 2304 (d) 2704
  8. 8.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68
  9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  10. 10.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 8
  11. 11.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
  12. 12.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 3 and 9 (c) 1 and 27 (d) 2 and 10
  13. 13.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
  14. 14.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 9604 (d) 10404
  15. 15.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 14 (d) 12
  16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 286 (d) 106
  17. 17.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 9 (d) 8
  18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  19. 19.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 256 (c) 196 (d) 376
  20. 20.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 8846 (d) 7396
  21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  22. 22.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
  24. 24.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 122 (c) 144 (d) 10
  25. 25.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 3 (d) 45
  26. 26.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 5 (c) 1 and 8 (d) 2 and 5
  27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  28. 28.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 9
  29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 277 (d) 529
  30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  31. 31.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 9 (c) 21 (d) 10
  32. 32.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9025 (c) 8125 (d) 9035
  33. 33.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 181 (c) 313 (d) 225
  34. 34.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 16 (d) 4
  35. 35.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 9 (d) 15
  36. 36.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
  37. 37.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 33 (c) 12 (d) 32
  38. 38.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 9 (d) 25
  39. 39.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
  40. 40.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)

Answer key

  1. 16
  2. 14
  3. (x + 8)(x + 9)
  4. (x + 3)(x + 8)
  5. 8 and 9
  6. (x + 7)(x + 9)
  7. 2304
  8. 68
  9. (x + 4)(x + 9)
  10. 3 and 6
  11. 4
  12. 3 and 9
  13. 9
  14. 10404
  15. 13
  16. 106
  17. 9
  18. (x + 6)(x + 7)
  19. 136
  20. 8836
  21. (x + 2)(x + 8)
  22. (x + 2)(x + 5)
  23. 5776
  24. 100
  25. 9
  26. 2 and 4
  27. 576
  28. 8
  29. 193
  30. (x + 6)(x + 8)
  31. 10
  32. 9025
  33. 137
  34. 16
  35. 8
  36. 65
  37. 32
  38. 9
  39. (x + 4)(x + 8)
  40. (x + 6)(x + 9)

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