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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 918 · 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 = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 244 (d) 484
  2. 2.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3721 (d) 3491
  3. 3.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 676 (d) 1156
  4. 4.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  5. 5.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 289 (d) 217
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  7. 7.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8846 (c) 8116 (d) 7396
  9. 9.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
  10. 10.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 7 (c) 1 and 15 (d) 2 and 6
  11. 11.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
  12. 12.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 875 (c) 1225 (d) 635
  13. 13.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  14. 14.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 65 (c) 73 (d) 81
  15. 15.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
  16. 16.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 5476 (d) 4916
  17. 17.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 10 (d) 13
  18. 18.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)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 5929 (d) 6889
  20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  21. 21.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 100 (d) 52
  22. 22.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 13 (d) 37
  23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  24. 24.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 90 (d) 198
  25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  26. 26.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 49 (c) 85 (d) 7
  27. 27.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 48 (d) 13
  28. 28.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 1 and 9 (d) 3 and 7
  29. 29.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 8 (d) 12
  30. 30.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
  31. 31.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  33. 33.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 64 (c) 50 (d) 78
  34. 34.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8649 (c) 8109 (d) 8659
  36. 36.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
  37. 37.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8099 (c) 8281 (d) 7931
  38. 38.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 166 (c) 144 (d) 122
  39. 39.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
  40. 40.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15

Answer key

  1. 164
  2. 3481
  3. 676
  4. (x + 2)(x + 5)
  5. 1
  6. (x + 8)(x + 9)
  7. (x + 3)(x + 8)
  8. 8836
  9. 3969
  10. 3 and 5
  11. 841
  12. 625
  13. 193
  14. 65
  15. 4
  16. 5476
  17. 10
  18. (x + 6)(x + 7)
  19. 6889
  20. (x + 2)(x + 3)
  21. 52
  22. 13
  23. (x + 7)(x + 8)
  24. 90
  25. (x + 7)(x + 9)
  26. 49
  27. 14
  28. 3 and 7
  29. 8
  30. 54
  31. 16
  32. (x + 6)(x + 9)
  33. 50
  34. (x + 3)(x + 5)
  35. 8649
  36. 961
  37. 7921
  38. 122
  39. 36
  40. 56

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