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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 822 · 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 = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 324 (d) 180
  2. 2.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 3 and 7 (c) 1 and 9 (d) 1 and 21
  3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 7235 (d) 5625
  5. 5.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 117 (d) 171
  6. 6.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 13 (d) 41
  7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  8. 8.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 17 (d) 19
  9. 9.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 3 (d) 9
  10. 10.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 122 (c) 144 (d) 10
  11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3609 (d) 3249
  12. 12.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 10
  13. 13.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 404 (d) 484
  14. 14.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  16. 16.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 9 (d) 81
  17. 17.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 3 and 5 (d) 2 and 6
  18. 18.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 29 (c) 28 (d) 27
  19. 19.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
  20. 20.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  23. 23.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  24. 24.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7396 (c) 8084 (d) 7406
  25. 25.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 6 (d) 36
  26. 26.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  27. 27.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 400 (c) 208 (d) 16
  28. 28.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 289 (d) 529
  29. 29.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 3 and 6 (c) 2 and 7 (d) 1 and 8
  30. 30.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
  31. 31.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 89 (c) 3 (d) 9
  32. 32.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 4916 (d) 5486
  33. 33.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  34. 34.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
  35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
  36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  37. 37.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 253 (d) 85
  38. 38.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096
  39. 39.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
  40. 40.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208

Answer key

  1. 180
  2. 3 and 7
  3. (x + 3)(x + 9)
  4. 7225
  5. 117
  6. 42
  7. (x + 8)(x + 9)
  8. 18
  9. 9
  10. 100
  11. 3969
  12. 24
  13. 484
  14. (x + 2)(x + 6)
  15. 17
  16. 81
  17. 3 and 5
  18. 28
  19. 5 and 7
  20. (x + 2)(x + 7)
  21. (x + 6)(x + 8)
  22. (x + 7)(x + 9)
  23. 20
  24. 7396
  25. 36
  26. (x + 3)(x + 8)
  27. 16
  28. 529
  29. 2 and 7
  30. 185
  31. 9
  32. 5476
  33. 72
  34. 4
  35. 13
  36. (x + 5)(x + 6)
  37. 85
  38. 4096
  39. 5041
  40. 160

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