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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 844 · 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 = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
  2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15
  4. 4.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 5 (c) 49 (d) 25
  5. 5.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 142 (d) 58
  6. 6.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  7. 7.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 12 (c) 14 (d) 13
  8. 8.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 657 (d) 333
  9. 9.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 9409 (d) 10619
  10. 10.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 5 and 6 (c) 1 and 30 (d) 2 and 15
  11. 11.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  13. 13.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  14. 14.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 73 (c) 81 (d) 65
  15. 15.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
  16. 16.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
  17. 17.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
  18. 18.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 625 (d) 225
  20. 20.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
  21. 21.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 8 (d) 6
  22. 22.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 4 (d) 16
  23. 23.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 9 (d) 11
  25. 25.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 5625 (c) 5635 (d) 4925
  26. 26.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2491 (d) 2219
  27. 27.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 256 (d) 416
  29. 29.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  30. 30.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 7 (d) 49
  31. 31.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
  32. 32.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
  33. 33.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 3 and 6 (d) 1 and 18
  34. 34.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 256 (c) 8 (d) 160
  35. 35.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 169 (c) 147 (d) 125
  36. 36.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1235 (d) 1225
  37. 37.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 28 (d) 12
  38. 38.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)
  39. 39.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 20 (d) 52
  40. 40.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599

Answer key

  1. 137
  2. (x + 7)(x + 8)
  3. 56
  4. 25
  5. 58
  6. (x + 6)(x + 7)
  7. 13
  8. 225
  9. 10609
  10. 5 and 6
  11. 1936
  12. (x + 8)(x + 9)
  13. (x + 6)(x + 9)
  14. 65
  15. 1024
  16. 1
  17. 676
  18. (x + 2)(x + 4)
  19. 225
  20. 7
  21. 6
  22. 16
  23. 841
  24. 10
  25. 5625
  26. 2209
  27. (x + 4)(x + 5)
  28. 576
  29. (x + 3)(x + 4)
  30. 49
  31. 15
  32. 48
  33. 3 and 6
  34. 64
  35. 125
  36. 1225
  37. 11
  38. (x + 5)(x + 9)
  39. 20
  40. 1521

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