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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 989 · 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 = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 256 (d) 352
  2. 2.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
  3. 3.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 101 (c) 121 (d) 9
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8474 (c) 8104 (d) 7744
  7. 7.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 5 and 6 (d) 1 and 10
  8. 8.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7921 (c) 7931 (d) 8281
  9. 9.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  11. 11.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  12. 12.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3364 (d) 3844
  13. 13.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  14. 14.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 5 (d) 8
  15. 15.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
  16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  17. 17.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 11 (d) 10
  18. 18.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
  19. 19.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 149 (d) 9
  20. 20.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2209 (d) 2491
  21. 21.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 3 and 8 (d) 1 and 24
  22. 22.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 19 (c) 13 (d) 1
  23. 23.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 14 (d) 48
  24. 24.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
  25. 25.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 11025 (d) 9975
  26. 26.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 289 (d) 223
  27. 27.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 7569 (d) 8109
  28. 28.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  29. 29.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 5 (d) 7
  30. 30.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
  31. 31.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 229 (d) 409
  32. 32.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 85 (c) 103 (d) 121
  33. 33.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2304 (d) 2704
  34. 34.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  35. 35.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  36. 36.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 429 (d) 219
  37. 37.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 6425 (d) 7225
  38. 38.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)
  39. 39.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 45 (d) 13
  40. 40.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 10 (c) 1 and 18 (d) 2 and 9

Answer key

  1. 160
  2. 8 and 9
  3. 81
  4. (x + 8)(x + 9)
  5. (x + 7)(x + 9)
  6. 8464
  7. 5 and 6
  8. 7921
  9. 8836
  10. (x + 7)(x + 8)
  11. 9
  12. 3364
  13. (x + 5)(x + 6)
  14. 6
  15. 164
  16. 17
  17. 10
  18. 170
  19. 9
  20. 2209
  21. 3 and 8
  22. 1
  23. 14
  24. 56
  25. 9025
  26. 157
  27. 8649
  28. (x + 4)(x + 5)
  29. 1
  30. 4 and 8
  31. 169
  32. 85
  33. 2704
  34. 181
  35. 34
  36. 149
  37. 7225
  38. (x + 2)(x + 3)
  39. 14
  40. 2 and 9

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