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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 861 · 40 questions
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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.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 657 (d) 333
  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.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)
  4. 4.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 5 (d) 193
  6. 6.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 14 (c) 16 (d) 8
  7. 7.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 3 and 8 (d) 1 and 24
  8. 8.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 4 (d) 106
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  10. 10.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
  12. 12.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 13 (d) 40
  13. 13.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
  14. 14.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 6 (c) 7 (d) 5
  15. 15.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 10 (d) 7
  16. 16.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
  17. 17.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 7 (d) 9
  18. 18.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 5 (c) 25 (d) 81
  19. 19.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 62 (d) 16
  20. 20.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 9 (c) 24 (d) 10
  21. 21.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 9 (c) 4 and 6 (d) 1 and 24
  22. 22.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  23. 23.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
  24. 24.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 8836 (d) 7396
  25. 25.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 39 (d) 49
  26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  27. 27.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  28. 28.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  29. 29.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1591 (d) 1849
  31. 31.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 81 (d) 45
  32. 32.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
  33. 33.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 11025 (d) 9025
  34. 34.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 592 (d) 208
  35. 35.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
  36. 36.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3854 (c) 3604 (d) 3364
  37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 884 (d) 676
  39. 39.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 125 (d) 225
  40. 40.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625

Answer key

  1. 225
  2. (x + 3)(x + 6)
  3. (x + 6)(x + 8)
  4. 8
  5. 25
  6. 15
  7. 3 and 8
  8. 16
  9. (x + 7)(x + 8)
  10. (x + 3)(x + 8)
  11. 10201
  12. 13
  13. 2916
  14. 6
  15. 7
  16. 28
  17. 8
  18. 25
  19. 63
  20. 10
  21. 4 and 6
  22. (x + 4)(x + 5)
  23. 11
  24. 8836
  25. 29
  26. 56
  27. 145
  28. (x + 2)(x + 8)
  29. (x + 3)(x + 4)
  30. 1369
  31. 45
  32. 7 and 9
  33. 9025
  34. 208
  35. 34
  36. 3844
  37. (x + 6)(x + 9)
  38. 676
  39. 25
  40. 625

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