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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 80 · 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 = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 34 (c) 2 (d) 64
  2. 2.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 5 (d) 25
  3. 3.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 3025 (d) 3625
  4. 4.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 146 (d) 366
  5. 5.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 9 (c) 1 and 24 (d) 4 and 6
  6. 6.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
  7. 7.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
  8. 8.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)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 739 (d) 1089
  10. 10.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 5 (d) 7
  11. 11.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 11 (c) 17 (d) 18
  12. 12.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1444 (d) 1596
  13. 13.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  14. 14.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 25 (c) 49 (d) 5
  15. 15.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 1625 (c) 2035 (d) 2025
  16. 16.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
  17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
  18. 18.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3721 (d) 3731
  19. 19.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1379 (d) 1849
  20. 20.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 113 (d) 337
  21. 21.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  22. 22.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
  23. 23.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 40 (c) 64 (d) 4
  24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  25. 25.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  26. 26.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 169 (c) 249 (d) 89
  27. 27.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1681 (c) 1521 (d) 1531
  28. 28.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 3 and 7 (c) 2 and 8 (d) 1 and 9
  29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  30. 30.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 8091 (d) 7579
  31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404
  32. 32.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
  33. 33.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 19 (d) 17
  34. 34.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  36. 36.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 13 (d) 15
  37. 37.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 36 (c) 37 (d) 13
  38. 38.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 74 (d) 214
  39. 39.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8101 (d) 8291

Answer key

  1. 4
  2. 25
  3. 4225
  4. 146
  5. 4 and 6
  6. 225
  7. 7 and 8
  8. (x + 2)(x + 4)
  9. 729
  10. 5
  11. 18
  12. 1444
  13. (x + 3)(x + 8)
  14. 25
  15. 2025
  16. 9
  17. 72
  18. 3721
  19. 1369
  20. 113
  21. 63
  22. 961
  23. 16
  24. 181
  25. (x + 4)(x + 8)
  26. 89
  27. 1521
  28. 2 and 8
  29. 17
  30. 7569
  31. 9604
  32. 4
  33. 18
  34. (x + 4)(x + 9)
  35. (x + 8)(x + 9)
  36. 14
  37. 36
  38. 74
  39. (x + 2)(x + 8)
  40. 8281

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