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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 158 · 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 − 10x + 24 = 0? (a) 24 (b) 10 (c) 25 (d) 23
  2. 2.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
  3. 3.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 166 (c) 122 (d) 144
  4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  5. 5.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)
  6. 6.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 382 (d) 130
  8. 8.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 3 (c) 9 (d) 65
  9. 9.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 2 and 6 (d) 3 and 5
  10. 10.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 9604 (d) 10404
  11. 11.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6734 (d) 6724
  12. 12.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
  13. 13.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 4 (c) 100 (d) 16
  14. 14.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
  15. 15.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
  16. 16.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
  17. 17.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 10 (c) 9 (d) 8
  18. 18.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 145 (c) 169 (d) 11
  19. 19.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)
  20. 20.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 25 (d) 5
  21. 21.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 170 (d) 478
  22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
  23. 23.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 729 (c) 1089 (d) 739
  24. 24.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)
  25. 25.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  26. 26.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 23 (c) 25 (d) 11
  27. 27.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 4 (c) 136 (d) 16
  28. 28.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  29. 29.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
  30. 30.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
  31. 31.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
  32. 32.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
  33. 33.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9025 (d) 9035
  34. 34.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 85 (c) 157 (d) 121
  35. 35.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 6 (d) 12
  36. 36.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 5625 (d) 4225
  37. 37.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7
  38. 38.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 5329 (c) 4489 (d) 4499
  39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  40. 40.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9

Answer key

  1. 24
  2. 2 and 9
  3. 122
  4. (x + 4)(x + 8)
  5. (x + 2)(x + 4)
  6. 34
  7. 130
  8. 9
  9. 3 and 5
  10. 10404
  11. 6724
  12. 16
  13. 16
  14. 1
  15. 5625
  16. 193
  17. 9
  18. 121
  19. (x + 6)(x + 7)
  20. 25
  21. 170
  22. 484
  23. 729
  24. (x + 7)(x + 8)
  25. (x + 4)(x + 7)
  26. 24
  27. 16
  28. (x + 2)(x + 6)
  29. (x + 4)(x + 6)
  30. 16
  31. 15
  32. 10201
  33. 9025
  34. 85
  35. 7
  36. 5625
  37. 6 and 7
  38. 4489
  39. (x + 5)(x + 8)
  40. 6 and 9

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