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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 525 · 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 = 54, what is a² + b²? (a) 171 (b) 117 (c) 333 (d) 225
  2. 2.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1601 (c) 1521 (d) 1691
  3. 3.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2809 (d) 2509
  4. 4.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 10009 (d) 9409
  5. 5.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
  6. 6.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 6 and 7 (d) 1 and 12
  7. 7.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  8. 8.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 13 (d) 42
  9. 9.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3249 (d) 3969
  10. 10.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 2 and 10 (c) 4 and 5 (d) 1 and 8
  11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
  12. 12.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
  13. 13.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 361 (d) 529
  14. 14.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 81 (c) 153 (d) 9
  15. 15.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
  16. 16.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
  17. 17.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 4 (c) 16 (d) 144
  18. 18.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 40 (d) 39
  19. 19.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5929 (d) 5939
  20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  22. 22.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 4225 (c) 5625 (d) 5635
  23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  24. 24.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 4 (d) 100
  25. 25.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2209 (d) 2809
  26. 26.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
  27. 27.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 24 (d) 25
  28. 28.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
  29. 29.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
  30. 30.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8116 (c) 8846 (d) 7396
  31. 31.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 12 (d) 11
  32. 32.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
  33. 33.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1774 (d) 1444
  34. 34.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 46 (d) 45
  35. 35.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 121 (d) 61
  36. 36.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9996 (d) 9614
  37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 225 (d) 235
  39. 39.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 11 (d) 24
  40. 40.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3854 (d) 3364

Answer key

  1. 117
  2. 1681
  3. 2809
  4. 10609
  5. (x + 2)(x + 9)
  6. 6 and 7
  7. 72
  8. 13
  9. 3969
  10. 4 and 5
  11. (x + 5)(x + 6)
  12. 5 and 6
  13. 193
  14. 81
  15. 1
  16. (x + 3)(x + 9)
  17. 16
  18. 40
  19. 5929
  20. (x + 5)(x + 8)
  21. 17
  22. 5625
  23. (x + 6)(x + 7)
  24. 4
  25. 2209
  26. 30
  27. 24
  28. 130
  29. 45
  30. 8836
  31. 12
  32. 4 and 7
  33. 1764
  34. 45
  35. 61
  36. 9604
  37. (x + 2)(x + 7)
  38. 225
  39. 11
  40. 3844

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