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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 619 · 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 68²? (a) 4896 (b) 4634 (c) 4624 (d) 5184
  2. 2.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  3. 3.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 2 (c) 100 (d) 4
  4. 4.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2401 (d) 2611
  5. 5.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 12 (d) 1 and 40
  6. 6.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 11 (d) 13
  7. 7.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  8. 8.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 1 and 9 (d) 3 and 7
  9. 9.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
  10. 10.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 1 and 24 (d) 2 and 12
  11. 11.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
  12. 12.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
  13. 13.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 69 (c) 49 (d) 29
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 9 (d) 20
  16. 16.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 324 (d) 484
  17. 17.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  18. 18.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  19. 19.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 361 (d) 181
  20. 20.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 16 (d) 54
  21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 42 (d) 14
  22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744
  23. 23.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 153 (d) 81
  24. 24.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7
  25. 25.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  26. 26.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 21 (d) 20
  27. 27.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  28. 28.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 4 and 5 (c) 1 and 8 (d) 1 and 20
  29. 29.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)
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  31. 31.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
  32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1369 (d) 1591
  33. 33.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2411 (d) 2401
  34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 72 (d) 17
  35. 35.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 24 (d) 9
  36. 36.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 8125 (c) 9025 (d) 7225
  37. 37.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 106 (c) 286 (d) 151
  38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
  39. 39.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4225 (c) 4235 (d) 4875
  40. 40.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)

Answer key

  1. 4624
  2. (x + 4)(x + 8)
  3. 4
  4. 2601
  5. 5 and 8
  6. 12
  7. (x + 2)(x + 8)
  8. 3 and 7
  9. 225
  10. 3 and 8
  11. 12
  12. 1089
  13. 29
  14. (x + 7)(x + 8)
  15. 9
  16. 324
  17. 15
  18. (x + 3)(x + 6)
  19. 1
  20. 15
  21. 13
  22. 8464
  23. 81
  24. 8
  25. 42
  26. 21
  27. (x + 3)(x + 4)
  28. 4 and 5
  29. (x + 8)(x + 9)
  30. 17
  31. 8 and 9
  32. 1369
  33. 2401
  34. 72
  35. 10
  36. 9025
  37. 106
  38. 185
  39. 4225
  40. (x + 4)(x + 9)

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