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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 279 · 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 = 10, what is a² + b²? (a) 29 (b) 69 (c) 39 (d) 49
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  3. 3.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
  5. 5.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
  6. 6.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
  7. 7.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9999 (d) 9811
  8. 8.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55
  9. 9.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 17 (c) 18 (d) 11
  10. 10.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 8 (d) 64
  11. 11.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)
  12. 12.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 163 (d) 262
  13. 13.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 5051 (d) 4761
  14. 14.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 361 (d) 441
  15. 15.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 36 (c) 6 (d) 256
  16. 16.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  17. 17.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 2 and 24 (d) 6 and 8
  18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 9801 (d) 10211
  20. 20.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  21. 21.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 202 (d) 4
  22. 22.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 136 (c) 4 (d) 256
  23. 23.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 10004 (d) 9604
  24. 24.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 625 (d) 925
  25. 25.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 9 (d) 3
  26. 26.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 6 and 9 (d) 1 and 54
  27. 27.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 9 (d) 3
  28. 28.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 65 (c) 7 (d) 81
  29. 29.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
  30. 30.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
  32. 32.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  33. 33.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 4
  34. 34.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 43 (d) 41
  35. 35.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 1 and 28 (c) 2 and 14 (d) 4 and 7
  36. 36.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 9 (c) 81 (d) 101
  37. 37.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2491 (c) 2209 (d) 2219
  38. 38.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 14 (d) 48
  39. 39.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 17 (d) 71
  40. 40.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)

Answer key

  1. 29
  2. (x + 8)(x + 9)
  3. (x + 2)(x + 4)
  4. 6241
  5. 15
  6. 441
  7. 9801
  8. 54
  9. 18
  10. 64
  11. (x + 7)(x + 9)
  12. 130
  13. 5041
  14. 361
  15. 36
  16. 1681
  17. 6 and 8
  18. (x + 6)(x + 8)
  19. 10201
  20. 8836
  21. 4
  22. 16
  23. 10404
  24. 1225
  25. 9
  26. 6 and 9
  27. 9
  28. 49
  29. 5625
  30. 181
  31. 4
  32. (x + 3)(x + 9)
  33. 5
  34. 42
  35. 4 and 7
  36. 81
  37. 2209
  38. 14
  39. 72
  40. (x + 3)(x + 8)

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