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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 913 · 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.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9216 (d) 9984
  3. 3.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 14 (c) 40 (d) 12
  4. 4.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 404 (d) 494
  5. 5.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4235 (d) 4225
  6. 6.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  7. 7.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 144 (c) 10 (d) 122
  8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  9. 9.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 193 (d) 529
  10. 10.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
  11. 11.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  12. 12.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 69 (d) 29
  13. 13.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 361 (d) 399
  14. 14.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
  15. 15.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 9 (d) 21
  16. 16.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
  17. 17.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 12 (d) 7
  18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 225 (c) 635 (d) 425
  20. 20.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 130 (d) 256
  21. 21.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  22. 22.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
  23. 23.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 11 (d) 10
  24. 24.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 333 (c) 171 (d) 225
  25. 25.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 64 (c) 160 (d) 8
  26. 26.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 2025 (d) 1225
  27. 27.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6399 (d) 6561
  28. 28.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 33 (d) 31
  29. 29.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  30. 30.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  31. 31.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 6 (c) 2 and 6 (d) 1 and 12
  32. 32.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 64 (c) 50 (d) 78
  33. 33.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 5929 (d) 6889
  35. 35.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
  36. 36.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 11 (d) 10
  37. 37.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 6 (c) 50 (d) 36
  38. 38.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 36 (c) 34 (d) 12
  39. 39.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 961 (d) 841
  40. 40.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 25 (d) 23

Answer key

  1. (x + 4)(x + 7)
  2. 9216
  3. 13
  4. 484
  5. 4225
  6. (x + 2)(x + 6)
  7. 100
  8. (x + 7)(x + 8)
  9. 193
  10. (x + 3)(x + 7)
  11. (x + 4)(x + 8)
  12. 29
  13. 361
  14. 2 and 8
  15. 10
  16. 15
  17. 8
  18. (x + 5)(x + 7)
  19. 625
  20. 4
  21. 1936
  22. 73
  23. 11
  24. 117
  25. 64
  26. 1225
  27. 6241
  28. 32
  29. 56
  30. (x + 3)(x + 9)
  31. 3 and 4
  32. 50
  33. (x + 3)(x + 4)
  34. 5929
  35. 2 and 6
  36. 11
  37. 36
  38. 35
  39. 961
  40. 24

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