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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 818 · 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 − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1946 (d) 1296
  5. 5.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 14 (d) 15
  6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  7. 7.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
  8. 8.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 16 (d) 4
  9. 9.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 16 (d) 11
  10. 10.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  11. 11.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 1 and 16 (c) 2 and 8 (d) 3 and 7
  12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  13. 13.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 1166 (c) 676 (d) 916
  15. 15.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8281 (d) 8291
  16. 16.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 429 (d) 149
  17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  18. 18.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 2 (d) 4
  19. 19.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  20. 20.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 225 (d) 337
  21. 21.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
  22. 22.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 56 (d) 14
  23. 23.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 49 (d) 25
  24. 24.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 24 (d) 10
  25. 25.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625
  26. 26.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
  27. 27.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 909 (d) 1099
  28. 28.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 4901 (d) 5051
  30. 30.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 297 (c) 225 (d) 153
  31. 31.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 7 (d) 49
  32. 32.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 9
  33. 33.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1379 (d) 1849
  34. 34.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 10
  35. 35.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
  36. 36.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 20 (d) 9
  37. 37.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 10609 (d) 9991
  38. 38.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 334 (d) 324
  39. 39.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 49 (d) 25
  40. 40.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404

Answer key

  1. 63
  2. (x + 8)(x + 9)
  3. (x + 7)(x + 9)
  4. 1936
  5. 15
  6. (x + 6)(x + 9)
  7. 4
  8. 16
  9. 10
  10. (x + 3)(x + 9)
  11. 2 and 8
  12. 56
  13. (x + 5)(x + 8)
  14. 1156
  15. 8281
  16. 149
  17. (x + 5)(x + 7)
  18. 4
  19. (x + 6)(x + 8)
  20. 113
  21. 5625
  22. 15
  23. 25
  24. 11
  25. 5625
  26. 8 and 9
  27. 1089
  28. (x + 2)(x + 6)
  29. 5041
  30. 153
  31. 49
  32. 18
  33. 1369
  34. 2 and 9
  35. 1
  36. 20
  37. 9409
  38. 324
  39. 1
  40. 484

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