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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 850 · 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.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 26 (d) 12
  2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 14 (d) 48
  3. 3.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 11 (c) 9 (d) 21
  4. 4.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
  5. 5.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 64 (d) 16
  6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 4106 (d) 3136
  8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 8101 (d) 7921
  9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  10. 10.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  12. 12.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 149 (d) 289
  13. 13.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 2 and 20 (c) 1 and 12 (d) 1 and 40
  14. 14.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 625 (d) 875
  15. 15.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  16. 16.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 32 (d) 13
  17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
  18. 18.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
  19. 19.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5635 (d) 5625
  20. 20.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2411 (d) 2601
  21. 21.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 289 (d) 157
  22. 22.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 106 (d) 196
  23. 23.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 81 (d) 41
  24. 24.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 7 (c) 225 (d) 137
  25. 25.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  26. 26.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
  27. 27.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1849 (c) 1591 (d) 1379
  28. 28.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
  29. 29.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
  30. 30.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 63 (d) 1 and 15
  31. 31.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 14 (d) 16
  32. 32.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
  33. 33.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)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 404 (d) 484
  35. 35.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2304 (d) 2704
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 117 (c) 63 (d) 81
  37. 37.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 148 (c) 196 (d) 100
  38. 38.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 2 (d) 10
  39. 39.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6409 (d) 6899
  40. 40.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6

Answer key

  1. 27
  2. 14
  3. 10
  4. (x + 4)(x + 7)
  5. 63
  6. (x + 7)(x + 9)
  7. 4096
  8. 8281
  9. (x + 3)(x + 4)
  10. (x + 5)(x + 9)
  11. 9
  12. 149
  13. 5 and 8
  14. 625
  15. (x + 6)(x + 7)
  16. 12
  17. 72
  18. 9801
  19. 5625
  20. 2401
  21. 157
  22. 16
  23. 41
  24. 49
  25. 9025
  26. 81
  27. 1369
  28. 14
  29. 1681
  30. 7 and 9
  31. 15
  32. (x + 3)(x + 8)
  33. (x + 5)(x + 8)
  34. 484
  35. 2704
  36. 45
  37. 100
  38. 4
  39. 6889
  40. 3 and 5

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