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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 339 · 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.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  2. 2.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 5 and 8 (c) 1 and 40 (d) 2 and 20
  3. 3.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  4. 4.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
  5. 5.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 1235 (d) 925
  6. 6.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 1 and 6 (d) 2 and 6
  7. 7.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  8. 8.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
  9. 9.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 30 (d) 11
  10. 10.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 409 (d) 169
  11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  12. 12.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 91 (c) 121 (d) 61
  13. 13.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180
  14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  16. 16.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 144 (c) 198 (d) 117
  17. 17.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  18. 18.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 54 (d) 53
  19. 19.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 17 (d) 9
  20. 20.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 9801 (d) 10211
  21. 21.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
  22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  23. 23.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  24. 24.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 136 (d) 376
  25. 25.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 9025 (d) 8075
  26. 26.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
  27. 27.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 26 (c) 28 (d) 12
  28. 28.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 5051 (d) 4901
  29. 29.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10816 (c) 9216 (d) 10826
  30. 30.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 169 (d) 49
  31. 31.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 904 (d) 784
  32. 32.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 11 (d) 12
  33. 33.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 12 (d) 35
  34. 34.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  35. 35.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 2 and 7 (d) 1 and 8
  36. 36.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 53 (c) 81 (d) 109
  37. 37.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 14 (d) 13
  38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 4225 (d) 3625
  40. 40.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 49 (d) 289

Answer key

  1. (x + 2)(x + 8)
  2. 5 and 8
  3. 41
  4. 13
  5. 1225
  6. 3 and 4
  7. 16
  8. 13
  9. 11
  10. 169
  11. (x + 2)(x + 4)
  12. 1
  13. 180
  14. (x + 5)(x + 6)
  15. (x + 6)(x + 8)
  16. 90
  17. (x + 5)(x + 9)
  18. 54
  19. 9
  20. 10201
  21. 1
  22. (x + 8)(x + 9)
  23. 15
  24. 136
  25. 7225
  26. 3844
  27. 27
  28. 5041
  29. 10816
  30. 49
  31. 1024
  32. 11
  33. 12
  34. 56
  35. 2 and 7
  36. 53
  37. 14
  38. (x + 4)(x + 5)
  39. 4225
  40. 49

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