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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 587 · 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 = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  4. 4.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)
  5. 5.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 256 (d) 146
  6. 6.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
  7. 7.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3249 (d) 3259
  8. 8.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2025 (d) 2035
  9. 9.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 4891 (d) 5329
  10. 10.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  11. 11.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
  12. 12.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  13. 13.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 49 (c) 69 (d) 39
  14. 14.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2516 (c) 2116 (d) 2916
  15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529
  16. 16.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
  17. 17.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 12 (d) 9
  18. 18.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  19. 19.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  20. 20.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 12 (c) 11 (d) 18
  21. 21.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
  22. 22.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 49 (d) 37
  23. 23.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
  24. 24.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 325 (d) 125
  25. 25.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 5939 (c) 6889 (d) 6391
  26. 26.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
  27. 27.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 47 (d) 14
  28. 28.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 8 (c) 9 (d) 10
  29. 29.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
  30. 30.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 144 (d) 74
  31. 31.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256
  32. 32.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
  33. 33.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 16 (d) 15
  34. 34.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
  35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4366 (d) 4356
  38. 38.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
  39. 39.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7744 (c) 8464 (d) 7754
  40. 40.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12

Answer key

  1. 185
  2. 17
  3. (x + 7)(x + 9)
  4. (x + 6)(x + 8)
  5. 146
  6. 193
  7. 3249
  8. 2025
  9. 4489
  10. (x + 6)(x + 9)
  11. 113
  12. 181
  13. 29
  14. 2916
  15. 529
  16. 170
  17. 8
  18. (x + 3)(x + 8)
  19. (x + 3)(x + 4)
  20. 11
  21. (x + 5)(x + 8)
  22. 37
  23. 4
  24. 125
  25. 5929
  26. 10816
  27. 48
  28. 9
  29. 10609
  30. 74
  31. 130
  32. 4
  33. 15
  34. 10
  35. (x + 5)(x + 7)
  36. (x + 7)(x + 8)
  37. 4356
  38. 5 and 6
  39. 7744
  40. 13

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