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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 113 · 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 37²? (a) 1591 (b) 1849 (c) 1379 (d) 1369
  2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  3. 3.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 85 (c) 169 (d) 127
  4. 4.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)
  5. 5.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 23 (c) 25 (d) 24
  6. 6.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 229 (c) 169 (d) 409
  7. 7.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 16 (d) 196
  8. 8.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
  9. 9.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 36 (d) 13
  10. 10.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 109 (d) 169
  11. 11.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  13. 13.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  14. 14.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 49 (c) 25 (d) 1
  15. 15.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 1 and 7 (d) 2 and 6
  16. 16.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 42 (d) 13
  17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3599 (d) 3481
  19. 19.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 144 (c) 117 (d) 198
  20. 20.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7
  21. 21.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 20 (d) 52
  22. 22.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 6375 (d) 7225
  23. 23.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
  24. 24.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 376 (c) 256 (d) 196
  25. 25.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 225 (c) 9 (d) 3
  26. 26.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3136 (d) 3616
  27. 27.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10
  28. 28.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 166 (d) 122
  29. 29.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
  30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  31. 31.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
  32. 32.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 2 and 5 (c) 1 and 10 (d) 3 and 4
  33. 33.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2525 (d) 2025
  34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 217 (d) 1
  35. 35.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  36. 36.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 45 (c) 1 and 13 (d) 5 and 9
  37. 37.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 3 (c) 169 (d) 9
  38. 38.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
  39. 39.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 48 (c) 14 (d) 49
  40. 40.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)

Answer key

  1. 1369
  2. (x + 6)(x + 7)
  3. 1
  4. (x + 5)(x + 9)
  5. 24
  6. 169
  7. 16
  8. 8
  9. 36
  10. 109
  11. (x + 3)(x + 6)
  12. 17
  13. 14
  14. 1
  15. 2 and 6
  16. 13
  17. (x + 8)(x + 9)
  18. 3481
  19. 90
  20. 7
  21. 20
  22. 5625
  23. 961
  24. 136
  25. 9
  26. 4096
  27. 11
  28. 122
  29. 10201
  30. (x + 3)(x + 7)
  31. 1
  32. 2 and 5
  33. 3025
  34. 1
  35. (x + 5)(x + 8)
  36. 5 and 9
  37. 9
  38. 6 and 7
  39. 48
  40. (x + 3)(x + 9)

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