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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 672 · 40 questions
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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 = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  2. 2.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561
  5. 5.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 9 (d) 15
  6. 6.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  8. 8.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 67 (c) 81 (d) 109
  9. 9.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2525 (d) 2025
  10. 10.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 13 (c) 1 and 48 (d) 2 and 24
  11. 11.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 2 and 7 (d) 3 and 6
  12. 12.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 45 (d) 9
  13. 13.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 6 and 9 (d) 1 and 54
  14. 14.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)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6409 (c) 5929 (d) 6899
  16. 16.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 7569 (c) 8649 (d) 8659
  17. 17.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 144 (c) 109 (d) 214
  18. 18.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
  19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101
  21. 21.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 64 (d) 88
  22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9999 (d) 9811
  23. 23.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 6 (d) 7
  24. 24.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9975 (c) 11025 (d) 9035
  25. 25.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 16 (d) 170
  26. 26.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 144 (d) 80
  27. 27.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 297 (c) 189 (d) 153
  28. 28.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  29. 29.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 304 (d) 208
  30. 30.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 1 and 15 (d) 3 and 5
  31. 31.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 10 (d) 11
  32. 32.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 139 (d) 229
  33. 33.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3979 (c) 3609 (d) 3249
  35. 35.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  36. 36.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  37. 37.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 5776 (d) 6416
  38. 38.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
  39. 39.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
  40. 40.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256

Answer key

  1. 61
  2. 256
  3. (x + 7)(x + 9)
  4. 6241
  5. 14
  6. (x + 2)(x + 3)
  7. (x + 5)(x + 8)
  8. 53
  9. 3025
  10. 6 and 8
  11. 2 and 7
  12. 9
  13. 6 and 9
  14. (x + 5)(x + 9)
  15. 6889
  16. 8649
  17. 74
  18. 2304
  19. (x + 3)(x + 7)
  20. 8281
  21. 40
  22. 9801
  23. 8
  24. 9025
  25. 16
  26. 80
  27. 153
  28. 15
  29. 208
  30. 3 and 5
  31. 11
  32. 109
  33. (x + 6)(x + 9)
  34. 3969
  35. (x + 5)(x + 6)
  36. 181
  37. 7056
  38. 4
  39. 1
  40. 36

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