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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 596 · 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 − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
  2. 2.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)
  3. 3.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9991 (c) 9419 (d) 9409
  4. 4.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
  5. 5.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
  6. 6.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 241 (d) 97
  7. 7.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 225 (d) 333
  8. 8.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)
  9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  10. 10.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 8 (c) 144 (d) 104
  11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
  12. 12.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 8099 (d) 7931
  13. 13.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 64 (d) 16
  14. 14.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
  15. 15.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4225 (c) 4925 (d) 5625
  16. 16.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
  17. 17.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
  18. 18.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 164 (c) 2 (d) 324
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  20. 20.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 36 (d) 34
  21. 21.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 4 (c) 16 (d) 64
  22. 22.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 146 (c) 256 (d) 366
  23. 23.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6409 (d) 6889
  24. 24.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 5184 (c) 4896 (d) 4634
  25. 25.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2926 (d) 2516
  26. 26.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 784 (d) 1024
  27. 27.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 8 (d) 7
  28. 28.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2496 (c) 2304 (d) 2704
  29. 29.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 100 (d) 36
  30. 30.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 8 (c) 6 (d) 12
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  33. 33.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  34. 34.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  36. 36.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)
  37. 37.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
  38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
  39. 39.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 1 and 14 (d) 7 and 8
  40. 40.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 36 (c) 6 (d) 324

Answer key

  1. 48
  2. (x + 3)(x + 6)
  3. 9409
  4. 40
  5. (x + 3)(x + 4)
  6. 97
  7. 117
  8. (x + 5)(x + 7)
  9. (x + 7)(x + 9)
  10. 64
  11. 72
  12. 7921
  13. 63
  14. 6 and 7
  15. 5625
  16. 7396
  17. 1156
  18. 4
  19. (x + 2)(x + 6)
  20. 35
  21. 16
  22. 146
  23. 6889
  24. 4624
  25. 2916
  26. 784
  27. 7
  28. 2304
  29. 36
  30. 7
  31. 1
  32. 17
  33. 6
  34. (x + 6)(x + 7)
  35. (x + 8)(x + 9)
  36. (x + 3)(x + 5)
  37. (x + 5)(x + 8)
  38. 16
  39. 7 and 8
  40. 36

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