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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 63 · 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.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 5329 (c) 4489 (d) 4499
  2. 2.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  3. 3.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 225 (d) 625
  4. 4.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  5. 5.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3625 (d) 3025
  6. 6.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 112 (c) 208 (d) 144
  7. 7.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
  8. 8.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 16 (d) 208
  9. 9.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  10. 10.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  12. 12.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 169 (c) 9 (d) 125
  13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  15. 15.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  16. 16.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
  17. 17.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 292 (c) 196 (d) 100
  18. 18.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  19. 19.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 10609 (d) 9409
  20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  21. 21.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 1 and 14 (d) 2 and 7
  22. 22.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 5 (d) 25
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 71 (d) 73
  24. 24.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 64 (c) 94 (d) 49
  25. 25.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1604 (d) 1444
  26. 26.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3364 (c) 3374 (d) 3844
  28. 28.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 31 (d) 30
  29. 29.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 100 (c) 142 (d) 79
  30. 30.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  31. 31.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  32. 32.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 13 (c) 5 and 9 (d) 1 and 45
  33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  34. 34.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
  35. 35.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
  36. 36.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  37. 37.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 24 (d) 11
  38. 38.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 16 (d) 10
  39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1599 (d) 1531
  40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)

Answer key

  1. 4489
  2. 676
  3. 225
  4. 113
  5. 4225
  6. 80
  7. 6889
  8. 16
  9. 1
  10. (x + 3)(x + 9)
  11. 17
  12. 81
  13. (x + 6)(x + 7)
  14. (x + 7)(x + 8)
  15. 15
  16. (x + 4)(x + 8)
  17. 100
  18. 484
  19. 9409
  20. (x + 5)(x + 8)
  21. 2 and 7
  22. 25
  23. 72
  24. 34
  25. 1764
  26. (x + 2)(x + 5)
  27. 3364
  28. 30
  29. 58
  30. 1
  31. (x + 3)(x + 8)
  32. 5 and 9
  33. (x + 3)(x + 7)
  34. 5 and 6
  35. 20
  36. (x + 6)(x + 9)
  37. 10
  38. 10
  39. 1521
  40. (x + 7)(x + 9)

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