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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 983 · 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 sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 4 (d) 6
  2. 2.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  3. 3.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 130 (d) 193
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 56 (d) 14
  6. 6.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
  7. 7.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  8. 8.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 10 (d) 4
  9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10609 (d) 10009
  11. 11.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6561 (d) 6241
  12. 12.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 8109 (d) 7569
  13. 13.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2116 (d) 2916
  14. 14.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 11 (d) 17
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  16. 16.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 7 (c) 65 (d) 81
  17. 17.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 13 (c) 10 (d) 16
  18. 18.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2601 (d) 2401
  19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  20. 20.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)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 9025 (c) 7235 (d) 8075
  22. 22.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  24. 24.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 7396 (d) 8836
  25. 25.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9216 (d) 9226
  26. 26.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 7
  27. 27.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  28. 28.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
  29. 29.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 1225 (d) 2035
  30. 30.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  31. 31.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 2 and 7 (d) 1 and 8
  32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1235 (c) 1225 (d) 925
  33. 33.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 73 (c) 5 (d) 121
  34. 34.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7
  35. 35.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 11 (d) 12
  36. 36.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601
  37. 37.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  38. 38.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 10 (c) 9 (d) 7
  39. 39.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 3136 (d) 4106
  40. 40.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 7 (d) 15

Answer key

  1. 5
  2. 16
  3. 130
  4. 17
  5. 15
  6. 13
  7. (x + 4)(x + 5)
  8. 4
  9. (x + 4)(x + 6)
  10. 10609
  11. 6241
  12. 8649
  13. 2916
  14. 18
  15. (x + 8)(x + 9)
  16. 49
  17. 10
  18. 2401
  19. 48
  20. (x + 5)(x + 9)
  21. 7225
  22. 145
  23. 72
  24. 8836
  25. 9216
  26. 12
  27. (x + 3)(x + 7)
  28. 28
  29. 2025
  30. 157
  31. 2 and 7
  32. 1225
  33. 25
  34. 6 and 7
  35. 11
  36. 3721
  37. 56
  38. 10
  39. 4096
  40. 8

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