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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 390 · 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 86²? (a) 7406 (b) 8836 (c) 7396 (d) 8084
  2. 2.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 625 (d) 635
  3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  5. 5.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
  6. 6.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 5329 (c) 4891 (d) 4499
  8. 8.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  10. 10.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 13 (c) 5 (d) 9
  11. 11.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 8125 (d) 9025
  12. 12.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 292 (d) 196
  13. 13.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 235 (d) 225
  14. 14.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 1089 (d) 909
  15. 15.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
  18. 18.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
  19. 19.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 3 and 6 (c) 2 and 7 (d) 1 and 8
  20. 20.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  21. 21.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
  22. 22.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 4 (d) 6
  23. 23.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 7754 (d) 8096
  24. 24.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
  25. 25.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8846 (d) 8836
  26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 382 (d) 193
  27. 27.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
  28. 28.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
  29. 29.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)
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 14 (d) 16
  31. 31.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 324 (d) 396
  32. 32.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
  34. 34.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 3 (c) 49 (d) 29
  35. 35.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 127 (d) 85
  36. 36.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
  37. 37.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
  38. 38.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  39. 39.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
  40. 40.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225

Answer key

  1. 7396
  2. 625
  3. 56
  4. (x + 8)(x + 9)
  5. 8 and 9
  6. (x + 3)(x + 6)
  7. 4489
  8. (x + 2)(x + 4)
  9. (x + 4)(x + 6)
  10. 5
  11. 9025
  12. 100
  13. 225
  14. 1089
  15. 181
  16. (x + 3)(x + 7)
  17. 1369
  18. 16
  19. 2 and 7
  20. (x + 6)(x + 8)
  21. 7 and 8
  22. 5
  23. 7744
  24. 4
  25. 8836
  26. 130
  27. 30
  28. 17
  29. (x + 4)(x + 5)
  30. 15
  31. 324
  32. (x + 2)(x + 8)
  33. 5041
  34. 9
  35. 85
  36. 36
  37. 2601
  38. 1
  39. 202
  40. 9

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