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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 898 · 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 product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 57 (d) 55
  2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  4. 4.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 5 and 9 (d) 1 and 45
  5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  6. 6.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 8 (c) 1 and 14 (d) 3 and 6
  7. 7.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
  8. 8.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 5625 (d) 6375
  9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
  10. 10.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 141 (d) 101
  11. 11.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3481 (c) 3721 (d) 3599
  12. 12.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 61 (d) 41
  13. 13.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 193 (d) 382
  14. 14.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 28 (c) 12 (d) 10
  15. 15.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 7 (c) 81 (d) 49
  16. 16.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
  17. 17.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 19 (d) 1
  18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 91 (d) 121
  19. 19.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 5 (c) 7 (d) 1
  20. 20.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 45 (d) 63
  21. 21.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2516 (c) 2926 (d) 2116
  22. 22.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 10 (d) 8
  23. 23.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 25 (c) 17 (d) 3
  24. 24.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10609 (d) 10619
  25. 25.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
  26. 26.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 1 (d) 85
  27. 27.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 93 (d) 177
  28. 28.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 85 (c) 103 (d) 121
  29. 29.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9025 (d) 9975
  30. 30.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
  31. 31.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
  32. 32.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 12 (d) 8
  34. 34.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
  35. 35.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
  36. 36.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 133 (c) 241 (d) 169
  37. 37.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 8 (c) 15 (d) 16
  38. 38.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
  39. 39.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 3 (c) 121 (d) 65
  40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17

Answer key

  1. 56
  2. (x + 2)(x + 4)
  3. (x + 4)(x + 9)
  4. 5 and 9
  5. 16
  6. 2 and 7
  7. 16
  8. 5625
  9. (x + 5)(x + 6)
  10. 101
  11. 3481
  12. 1
  13. 130
  14. 11
  15. 49
  16. 1
  17. 1
  18. 61
  19. 1
  20. 45
  21. 2916
  22. 9
  23. 9
  24. 10609
  25. 8 and 9
  26. 1
  27. 65
  28. 85
  29. 9025
  30. 324
  31. 576
  32. (x + 8)(x + 9)
  33. 7
  34. 9
  35. 3025
  36. 97
  37. 15
  38. 42
  39. 9
  40. 17

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