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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 687 · 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 = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
  2. 2.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  3. 3.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 13 (d) 11
  4. 4.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
  5. 5.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
  6. 6.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 28 (d) 12
  7. 7.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 4356 (d) 5476
  8. 8.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 28 (d) 1 and 10
  9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  10. 10.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 8836 (c) 7396 (d) 8116
  12. 12.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
  13. 13.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
  14. 14.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 35 (c) 13 (d) 36
  15. 15.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 121 (d) 85
  16. 16.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 5625 (d) 4225
  17. 17.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
  18. 18.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 121 (d) 85
  19. 19.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
  20. 20.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 4096 (c) 3584 (d) 3146
  21. 21.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5041 (d) 5051
  22. 22.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
  23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  24. 24.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 121 (c) 177 (d) 93
  25. 25.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
  26. 26.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3844 (d) 3374
  28. 28.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 64 (c) 57 (d) 50
  29. 29.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
  30. 30.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  31. 31.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
  32. 32.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  33. 33.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 324 (d) 4
  34. 34.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)
  35. 35.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 39 (c) 29 (d) 49
  36. 36.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 2475 (d) 3025
  37. 37.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 9 (d) 29
  38. 38.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
  39. 39.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
  40. 40.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 8464 (d) 7744

Answer key

  1. 225
  2. (x + 6)(x + 8)
  3. 12
  4. 256
  5. 10201
  6. 11
  7. 5476
  8. 4 and 7
  9. (x + 5)(x + 9)
  10. (x + 4)(x + 9)
  11. 8836
  12. 6 and 8
  13. (x + 2)(x + 3)
  14. 36
  15. 49
  16. 5625
  17. 4356
  18. 85
  19. 82
  20. 3136
  21. 5041
  22. 676
  23. (x + 6)(x + 7)
  24. 65
  25. 25
  26. (x + 3)(x + 6)
  27. 3364
  28. 50
  29. 1089
  30. (x + 4)(x + 6)
  31. 56
  32. (x + 3)(x + 4)
  33. 16
  34. (x + 7)(x + 8)
  35. 29
  36. 2025
  37. 9
  38. 1681
  39. 6
  40. 8464

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