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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 530 · 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 = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 208 (d) 160
  2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
  3. 3.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 7 (d) 1
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  5. 5.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 5184 (d) 4624
  6. 6.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
  7. 7.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 30 (d) 1 and 10
  8. 8.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 13 (d) 12
  9. 9.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
  10. 10.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
  11. 11.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  12. 12.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  13. 13.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 82 (c) 8 (d) 64
  14. 14.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
  15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  17. 17.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 4 (d) 7
  18. 18.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 229 (c) 409 (d) 169
  19. 19.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
  20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 541 (d) 361
  21. 21.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 61 (c) 91 (d) 121
  22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 875 (d) 625
  24. 24.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
  25. 25.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 52
  26. 26.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  27. 27.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
  28. 28.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 14 (c) 8 (d) 16
  29. 29.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 2025 (d) 1225
  30. 30.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 276 (c) 116 (d) 196
  31. 31.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 160 (c) 256 (d) 8
  32. 32.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 635 (c) 425 (d) 225
  33. 33.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 81 (c) 97 (d) 73
  34. 34.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 16 (d) 4
  35. 35.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 28 (d) 29
  36. 36.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 196 (d) 2
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 361 (d) 185
  39. 39.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 185 (d) 361
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72

Answer key

  1. 160
  2. 1
  3. 1
  4. 17
  5. 4624
  6. 1681
  7. 5 and 6
  8. 12
  9. 5929
  10. 7569
  11. 42
  12. (x + 6)(x + 8)
  13. 64
  14. 6 and 8
  15. (x + 6)(x + 9)
  16. (x + 4)(x + 5)
  17. 5
  18. 169
  19. 193
  20. 181
  21. 1
  22. (x + 8)(x + 9)
  23. 625
  24. 9604
  25. 4
  26. (x + 5)(x + 9)
  27. 6 and 7
  28. 15
  29. 1225
  30. 116
  31. 64
  32. 625
  33. 65
  34. 16
  35. 28
  36. 4
  37. (x + 7)(x + 8)
  38. 185
  39. 9
  40. 72

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