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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 490 · 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 − 15x + 54 = 0? (a) 15 (b) 54 (c) 14 (d) 16
  2. 2.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 2 and 18 (c) 1 and 36 (d) 4 and 9
  3. 3.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)
  4. 4.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 11 (d) 31
  5. 5.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 21 (d) 20
  6. 6.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 15 (d) 16
  7. 7.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 11 (d) 12
  8. 8.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  10. 10.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 169 (d) 11
  11. 11.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 35 (d) 11
  12. 12.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10619 (d) 10609
  13. 13.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 54 (d) 15
  14. 14.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 15 (d) 8
  15. 15.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6734 (d) 6724
  16. 16.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 177 (c) 121 (d) 93
  17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2704 (d) 2714
  18. 18.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 8 (d) 3 and 6
  19. 19.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 125 (d) 225
  20. 20.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 4 (d) 2
  21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  23. 23.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 15 (d) 13
  24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  25. 25.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 10 (d) 9
  26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 193 (d) 256
  27. 27.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 25 (d) 33
  28. 28.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8836 (c) 7396 (d) 8846
  29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
  30. 30.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
  31. 31.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 118 (c) 91 (d) 100
  32. 32.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 23 (d) 25
  33. 33.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 1 and 9 (c) 2 and 8 (d) 3 and 7
  34. 34.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  35. 35.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2219 (d) 2491
  36. 36.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 12 (c) 8 (d) 7
  37. 37.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 42 (d) 13
  38. 38.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 3025 (d) 2525
  39. 39.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
  40. 40.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3969 (d) 3609

Answer key

  1. 15
  2. 4 and 9
  3. (x + 4)(x + 8)
  4. 30
  5. 21
  6. 16
  7. 11
  8. (x + 3)(x + 5)
  9. (x + 6)(x + 8)
  10. 121
  11. 12
  12. 10609
  13. 54
  14. 15
  15. 6724
  16. 65
  17. 2704
  18. 3 and 6
  19. 25
  20. 4
  21. (x + 8)(x + 9)
  22. (x + 7)(x + 9)
  23. 14
  24. 16
  25. 10
  26. 130
  27. 17
  28. 8836
  29. 193
  30. 4225
  31. 82
  32. 24
  33. 2 and 8
  34. (x + 2)(x + 3)
  35. 2209
  36. 8
  37. 13
  38. 3025
  39. 8 and 9
  40. 3969

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