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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 142 · 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 sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  2. 2.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 58 (d) 142
  3. 3.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 27 (d) 3 and 9
  4. 4.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  5. 5.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 9025 (d) 8075
  6. 6.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)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2926 (c) 2916 (d) 2516
  8. 8.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)
  9. 9.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
  10. 10.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 784 (d) 794
  11. 11.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6401 (d) 6571
  12. 12.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 297 (c) 189 (d) 225
  13. 13.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
  14. 14.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
  15. 15.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 180 (d) 252
  16. 16.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 441 (d) 401
  17. 17.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9991 (c) 9419 (d) 9409
  18. 18.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)
  19. 19.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 198 (c) 144 (d) 117
  20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
  22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  23. 23.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  24. 24.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
  25. 25.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 5625 (d) 4225
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
  27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  28. 28.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 8 (d) 6
  29. 29.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 14 (d) 13
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
  31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
  33. 33.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 2 (d) 100
  34. 34.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  35. 35.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  36. 36.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 2 and 14 (d) 1 and 28
  37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  38. 38.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  39. 39.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 10211 (d) 9801
  40. 40.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 2 and 10 (d) 5 and 7

Answer key

  1. 16
  2. 58
  3. 3 and 9
  4. 289
  5. 7225
  6. (x + 7)(x + 9)
  7. 2916
  8. (x + 8)(x + 9)
  9. 48
  10. 784
  11. 6561
  12. 153
  13. 30
  14. 841
  15. 180
  16. 441
  17. 9409
  18. (x + 6)(x + 8)
  19. 90
  20. (x + 6)(x + 7)
  21. 13
  22. (x + 7)(x + 8)
  23. 2 and 6
  24. 6 and 7
  25. 5625
  26. 181
  27. (x + 3)(x + 8)
  28. 7
  29. 14
  30. 15
  31. 1
  32. (x + 5)(x + 7)
  33. 4
  34. 2 and 3
  35. 56
  36. 4 and 7
  37. 225
  38. (x + 3)(x + 6)
  39. 10201
  40. 5 and 7

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