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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 934 · 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.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 9 (d) 4 and 6
  2. 2.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 163 (c) 262 (d) 196
  3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  4. 4.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
  5. 5.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 41 (c) 81 (d) 1
  6. 6.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 9 (d) 10
  7. 7.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 7 (d) 1 and 14
  8. 8.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
  9. 9.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
  11. 11.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1444 (c) 1454 (d) 1596
  12. 12.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 73 (d) 65
  13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 225 (d) 153
  14. 14.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  15. 15.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 1 (d) 7
  16. 16.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
  17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6404 (d) 6724
  18. 18.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
  19. 19.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 4 (c) 2 (d) 36
  20. 20.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
  21. 21.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 4624 (d) 5194
  23. 23.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1235 (d) 1575
  24. 24.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 100 (c) 68 (d) 132
  25. 25.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 2 and 14 (c) 4 and 7 (d) 1 and 10
  26. 26.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
  28. 28.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)
  29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
  31. 31.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 33 (d) 32
  32. 32.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 3025 (d) 2525
  33. 33.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
  34. 34.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 324 (c) 404 (d) 494
  35. 35.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1225 (c) 1235 (d) 925
  36. 36.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
  37. 37.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
  38. 38.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 49 (c) 61 (d) 37
  39. 39.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
  40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)

Answer key

  1. 4 and 6
  2. 130
  3. (x + 5)(x + 9)
  4. 12
  5. 1
  6. 10
  7. 2 and 7
  8. 1156
  9. 10
  10. 15
  11. 1444
  12. 65
  13. 81
  14. (x + 4)(x + 5)
  15. 1
  16. 63
  17. 6724
  18. 10201
  19. 4
  20. 130
  21. (x + 5)(x + 6)
  22. 5184
  23. 1225
  24. 68
  25. 4 and 7
  26. 6
  27. 225
  28. (x + 3)(x + 4)
  29. (x + 7)(x + 8)
  30. 529
  31. 32
  32. 3025
  33. 4
  34. 484
  35. 1225
  36. 185
  37. (x + 2)(x + 7)
  38. 37
  39. 1
  40. (x + 3)(x + 6)

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