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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 28 · 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 = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 421 (d) 289
  2. 2.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 2 and 12 (d) 1 and 24
  3. 3.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 229 (d) 109
  4. 4.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 81 (c) 9 (d) 121
  5. 5.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 11 (c) 13 (d) 12
  6. 6.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 625 (d) 1235
  7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  9. 9.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
  10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 5476 (d) 4366
  12. 12.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 16 (c) 8 (d) 15
  13. 13.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 14 (d) 15
  14. 14.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 2 and 21 (c) 1 and 42 (d) 1 and 12
  15. 15.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 85 (c) 157 (d) 121
  16. 16.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 20 (c) 4 (d) 36
  17. 17.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 31 (d) 29
  18. 18.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
  19. 19.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
  20. 20.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 256 (d) 193
  21. 21.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 164 (d) 244
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  23. 23.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 324 (d) 484
  24. 24.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2491 (d) 2809
  25. 25.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 14 (d) 15
  26. 26.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  27. 27.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  28. 28.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 34 (d) 2
  29. 29.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7921 (c) 8281 (d) 7931
  30. 30.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)
  31. 31.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 43 (d) 41
  32. 32.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 6416 (d) 7056
  34. 34.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 169 (d) 85
  35. 35.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 125 (c) 225 (d) 175
  36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  37. 37.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
  38. 38.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
  39. 39.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
  40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7235 (d) 7225

Answer key

  1. 157
  2. 3 and 8
  3. 109
  4. 81
  5. 12
  6. 1225
  7. (x + 2)(x + 3)
  8. (x + 7)(x + 8)
  9. 4
  10. (x + 4)(x + 5)
  11. 4356
  12. 15
  13. 14
  14. 6 and 7
  15. 85
  16. 4
  17. 30
  18. 1
  19. 8 and 9
  20. 130
  21. 164
  22. 17
  23. 324
  24. 2209
  25. 14
  26. 72
  27. (x + 3)(x + 5)
  28. 4
  29. 7921
  30. (x + 8)(x + 9)
  31. 42
  32. (x + 7)(x + 9)
  33. 7056
  34. 85
  35. 125
  36. (x + 6)(x + 8)
  37. 1
  38. 54
  39. 841
  40. 7225

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