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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 580 · 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 = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 196 (d) 130
  2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  3. 3.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
  4. 4.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 144 (c) 104 (d) 184
  5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
  7. 7.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
  8. 8.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 89 (c) 249 (d) 169
  9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  10. 10.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 13 (d) 32
  11. 11.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
  12. 12.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 36 (d) 20
  13. 13.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 729 (d) 1089
  14. 14.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 19 (c) 18 (d) 11
  15. 15.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
  16. 16.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3374 (d) 3844
  17. 17.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
  18. 18.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8464 (d) 8096
  19. 19.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3731 (d) 3481
  20. 20.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 4901 (d) 5041
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  22. 22.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 35 (d) 5 and 7
  23. 23.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 10 (d) 22
  24. 24.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11035 (d) 11025
  25. 25.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9975 (d) 9025
  26. 26.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
  27. 27.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 3 (d) 9
  28. 28.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 4
  29. 29.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 625 (d) 635
  30. 30.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 1225 (d) 2025
  31. 31.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 16 (d) 4
  32. 32.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37
  33. 33.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  34. 34.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 5 and 6 (c) 1 and 30 (d) 2 and 15
  35. 35.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 9 (c) 121 (d) 65
  36. 36.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 21 (d) 3 and 7
  37. 37.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  38. 38.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 30 (d) 11
  39. 39.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
  40. 40.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496

Answer key

  1. 64
  2. (x + 2)(x + 4)
  3. 8 and 9
  4. 104
  5. (x + 8)(x + 9)
  6. 5625
  7. (x + 3)(x + 6)
  8. 89
  9. (x + 4)(x + 5)
  10. 12
  11. 25
  12. 4
  13. 729
  14. 18
  15. 2 and 9
  16. 3364
  17. 13
  18. 7744
  19. 3721
  20. 5041
  21. 181
  22. 5 and 7
  23. 10
  24. 11025
  25. 9025
  26. 7 and 8
  27. 9
  28. 5
  29. 625
  30. 1225
  31. 4
  32. 1
  33. (x + 3)(x + 8)
  34. 5 and 6
  35. 9
  36. 3 and 7
  37. (x + 3)(x + 9)
  38. 30
  39. 7
  40. 2304

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