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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 589 · 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 is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 9 (c) 18 (d) 19
  2. 2.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 25 (d) 23
  3. 3.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
  4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  5. 5.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  7. 7.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 225 (d) 113
  8. 8.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  9. 9.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 1 and 10 (d) 2 and 9
  10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  12. 12.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3249 (d) 3969
  13. 13.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 12 (d) 3 and 4
  14. 14.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4225 (c) 3625 (d) 4235
  15. 15.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 100 (c) 148 (d) 196
  16. 16.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 29 (c) 27 (d) 28
  17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  18. 18.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)
  19. 19.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 24 (d) 23
  20. 20.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 225 (c) 9 (d) 3
  21. 21.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 157 (c) 121 (d) 103
  22. 22.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  23. 23.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 11035 (d) 10025
  24. 24.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 5476 (d) 4884
  25. 25.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 171 (d) 225
  26. 26.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
  27. 27.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  28. 28.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  29. 29.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 28 (d) 12
  30. 30.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1584 (d) 1296
  31. 31.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 64 (d) 88
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  33. 33.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 25 (c) 121 (d) 5
  34. 34.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 129 (c) 169 (d) 249
  35. 35.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 10 (c) 16 (d) 9
  36. 36.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
  37. 37.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  38. 38.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 324 (d) 484
  39. 39.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4891 (c) 4489 (d) 5329

Answer key

  1. 18
  2. 24
  3. 5 and 6
  4. (x + 7)(x + 9)
  5. (x + 2)(x + 8)
  6. 1681
  7. 113
  8. 8 and 9
  9. 2 and 9
  10. (x + 6)(x + 7)
  11. 1
  12. 3249
  13. 2 and 6
  14. 4225
  15. 100
  16. 28
  17. 2704
  18. (x + 6)(x + 8)
  19. 24
  20. 9
  21. 85
  22. 56
  23. 11025
  24. 4356
  25. 117
  26. 72
  27. (x + 8)(x + 9)
  28. (x + 4)(x + 7)
  29. 11
  30. 1296
  31. 40
  32. (x + 5)(x + 9)
  33. 25
  34. 89
  35. 10
  36. (x + 5)(x + 8)
  37. (x + 2)(x + 5)
  38. 164
  39. (x + 7)(x + 8)
  40. 4489

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