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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 246 · 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 = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 37 (d) 49
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1764 (d) 1604
  4. 4.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 9 (c) 11 (d) 10
  5. 5.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 6889 (d) 5929
  7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 1225 (d) 2035
  8. 8.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 196 (c) 100 (d) 292
  9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
  10. 10.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
  11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 1 (d) 361
  12. 12.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4925 (c) 4225 (d) 5635
  13. 13.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 11 (d) 10
  14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  15. 15.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 4 (c) 2 and 3 (d) 1 and 6
  16. 16.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 7 (c) 5 (d) 9
  17. 17.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
  18. 18.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 28 (d) 26
  19. 19.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 149 (d) 429
  20. 20.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 1 and 54 (d) 2 and 27
  21. 21.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
  22. 22.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3364 (c) 3596 (d) 3844
  23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
  24. 24.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 81 (c) 169 (d) 125
  25. 25.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 148 (c) 76 (d) 100
  26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
  27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 256 (d) 416
  28. 28.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
  29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  31. 31.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 4 (c) 2 (d) 74
  32. 32.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 29 (d) 28
  33. 33.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3584 (c) 3146 (d) 3136
  34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  35. 35.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)
  36. 36.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 2 (d) 164
  37. 37.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  38. 38.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2499 (c) 2601 (d) 2401
  39. 39.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 28 (c) 4 and 7 (d) 1 and 10
  40. 40.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 1 and 18 (d) 3 and 6

Answer key

  1. 37
  2. (x + 4)(x + 5)
  3. 1764
  4. 10
  5. (x + 4)(x + 9)
  6. 6889
  7. 2025
  8. 100
  9. 185
  10. 72
  11. 1
  12. 5625
  13. 11
  14. (x + 2)(x + 3)
  15. 2 and 3
  16. 1
  17. 193
  18. 27
  19. 149
  20. 6 and 9
  21. 7 and 9
  22. 3364
  23. (x + 6)(x + 9)
  24. 81
  25. 52
  26. 145
  27. 576
  28. 6084
  29. (x + 6)(x + 8)
  30. (x + 5)(x + 9)
  31. 4
  32. 28
  33. 3136
  34. (x + 5)(x + 6)
  35. (x + 7)(x + 9)
  36. 4
  37. 16
  38. 2401
  39. 4 and 7
  40. 3 and 6

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