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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 967 · 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.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 539 (d) 289
  2. 2.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 45 (d) 3
  3. 3.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 118 (c) 100 (d) 82
  4. 4.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1444 (c) 1774 (d) 1764
  5. 5.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 49 (d) 7
  6. 6.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
  7. 7.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 169 (d) 225
  8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  9. 9.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 85 (d) 169
  10. 10.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 1225 (c) 925 (d) 1235
  12. 12.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 334 (d) 324
  13. 13.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 16 (d) 54
  14. 14.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 9 (d) 12
  15. 15.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
  16. 16.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
  17. 17.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 7 (c) 3 and 5 (d) 1 and 15
  18. 18.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 43 (d) 42
  19. 19.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)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
  21. 21.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 208 (c) 16 (d) 400
  22. 22.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 7 (d) 9
  23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
  24. 24.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  25. 25.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 81 (c) 9 (d) 125
  26. 26.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 7 (c) 169 (d) 109
  27. 27.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 47 (d) 48
  28. 28.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
  29. 29.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 9 (d) 10
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  31. 31.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
  32. 32.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 10 (d) 21
  33. 33.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 5041 (d) 4761
  35. 35.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 117 (d) 225
  36. 36.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  38. 38.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 34 (d) 12
  39. 39.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 4 (d) 7
  40. 40.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 5 (d) 25

Answer key

  1. 529
  2. 9
  3. 82
  4. 1764
  5. 49
  6. 225
  7. 1
  8. (x + 6)(x + 9)
  9. 1
  10. (x + 4)(x + 8)
  11. 1225
  12. 324
  13. 15
  14. 8
  15. 14
  16. 16
  17. 3 and 5
  18. 42
  19. (x + 3)(x + 8)
  20. 3721
  21. 16
  22. 8
  23. 56
  24. (x + 2)(x + 6)
  25. 81
  26. 49
  27. 48
  28. 6 and 7
  29. 9
  30. (x + 7)(x + 8)
  31. 6 and 8
  32. 21
  33. 4
  34. 4761
  35. 117
  36. 1521
  37. (x + 4)(x + 7)
  38. 35
  39. 5
  40. 25

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