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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 865 · 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.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  2. 2.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  3. 3.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 16 (c) 80 (d) 144
  4. 4.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 185 (d) 361
  5. 5.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 7 (d) 5
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  7. 7.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 225 (d) 153
  8. 8.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 399 (d) 361
  9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 42 (d) 13
  10. 10.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 61 (d) 1
  11. 11.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 6
  12. 12.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7066 (c) 5776 (d) 7056
  13. 13.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 7056 (d) 5786
  14. 14.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 20 (d) 10
  15. 15.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
  16. 16.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 25 (d) 23
  17. 17.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 16 (d) 10
  18. 18.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 425 (d) 635
  19. 19.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 81 (c) 109 (d) 67
  20. 20.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 85 (d) 103
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
  22. 22.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 400 (d) 2
  23. 23.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
  24. 24.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 133 (c) 166 (d) 144
  25. 25.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 52
  26. 26.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
  27. 27.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 1034 (c) 784 (d) 904
  28. 28.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 34 (d) 49
  29. 29.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 177 (d) 65
  30. 30.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  31. 31.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10
  32. 32.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  33. 33.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 10 (c) 3 and 6 (d) 1 and 18
  34. 34.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
  35. 35.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 4 (d) 136
  36. 36.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 29 (c) 49 (d) 39
  37. 37.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
  38. 38.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 25 (c) 11 (d) 23
  39. 39.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
  40. 40.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149

Answer key

  1. (x + 3)(x + 9)
  2. (x + 2)(x + 8)
  3. 16
  4. 185
  5. 6
  6. (x + 7)(x + 8)
  7. 153
  8. 361
  9. 13
  10. 1
  11. 7
  12. 7056
  13. 5776
  14. 21
  15. 1936
  16. 24
  17. 10
  18. 625
  19. 53
  20. 85
  21. 181
  22. 4
  23. 54
  24. 122
  25. 4
  26. 208
  27. 1024
  28. 34
  29. 65
  30. (x + 4)(x + 8)
  31. 11
  32. (x + 3)(x + 8)
  33. 2 and 9
  34. 145
  35. 16
  36. 29
  37. 85
  38. 24
  39. 11
  40. 9

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