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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 133 · 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 − 9x + 20 = 0? (a) 8 (b) 20 (c) 9 (d) 10
  2. 2.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 100 (d) 58
  3. 3.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 9 (d) 1 and 18
  4. 4.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 1 and 13 (d) 6 and 8
  5. 5.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
  6. 6.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 41 (c) 40 (d) 13
  7. 7.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 193 (d) 382
  8. 8.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 106 (c) 196 (d) 4
  9. 9.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 185 (c) 3 (d) 9
  10. 10.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)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
  12. 12.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 9 (d) 225
  14. 14.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 7 (d) 8
  15. 15.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 100 (c) 196 (d) 148
  16. 16.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1849 (d) 1369
  17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  20. 20.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)
  21. 21.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  23. 23.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 28 (c) 29 (d) 27
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
  25. 25.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 7 (d) 15
  26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  28. 28.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
  29. 29.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)
  30. 30.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
  31. 31.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 256 (d) 266
  32. 32.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 36 (d) 12
  33. 33.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
  34. 34.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 32 (d) 33
  35. 35.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 2 and 4 (d) 1 and 4
  36. 36.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
  37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 416 (d) 256
  39. 39.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 10 (d) 24
  40. 40.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 30 (d) 31

Answer key

  1. 9
  2. 58
  3. 3 and 6
  4. 6 and 8
  5. (x + 3)(x + 7)
  6. 40
  7. 130
  8. 16
  9. 9
  10. (x + 2)(x + 6)
  11. 484
  12. (x + 7)(x + 9)
  13. 81
  14. 8
  15. 100
  16. 1849
  17. (x + 2)(x + 3)
  18. 4096
  19. (x + 8)(x + 9)
  20. (x + 5)(x + 7)
  21. (x + 6)(x + 9)
  22. (x + 6)(x + 7)
  23. 28
  24. (x + 3)(x + 6)
  25. 8
  26. (x + 4)(x + 7)
  27. (x + 4)(x + 8)
  28. 16
  29. (x + 5)(x + 8)
  30. 42
  31. 256
  32. 13
  33. 15
  34. 32
  35. 2 and 3
  36. 49
  37. (x + 2)(x + 4)
  38. 576
  39. 11
  40. 30

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