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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 41 · 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 = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
  2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  3. 3.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 130 (c) 2 (d) 256
  4. 4.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 7 (d) 1 and 12
  5. 5.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
  6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
  7. 7.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 1 and 20 (d) 4 and 5
  8. 8.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)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1601 (d) 1681
  10. 10.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)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 1225 (d) 875
  12. 12.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  13. 13.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 3 and 7 (c) 2 and 8 (d) 1 and 9
  14. 14.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 2 and 4 (d) 1 and 6
  15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
  17. 17.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
  18. 18.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2126 (d) 2916
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
  20. 20.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 4 and 6 (d) 2 and 12
  21. 21.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 625 (c) 635 (d) 225
  22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
  23. 23.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 52
  24. 24.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 80 (c) 16 (d) 4
  25. 25.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 169 (d) 113
  26. 26.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 5329 (d) 4891
  27. 27.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
  28. 28.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  29. 29.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 25 (d) 5
  30. 30.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 79 (d) 100
  31. 31.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 4909 (d) 5329
  32. 32.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
  33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  34. 34.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 20 (d) 4
  35. 35.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 54 (d) 53
  36. 36.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 11 (d) 30
  37. 37.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
  38. 38.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
  39. 39.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 85 (d) 169
  40. 40.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 10 (d) 30

Answer key

  1. 45
  2. 15
  3. 4
  4. 2 and 6
  5. 10609
  6. 72
  7. 4 and 5
  8. (x + 3)(x + 7)
  9. 1681
  10. (x + 6)(x + 7)
  11. 625
  12. (x + 3)(x + 4)
  13. 2 and 8
  14. 2 and 3
  15. (x + 5)(x + 8)
  16. 1225
  17. 208
  18. 2116
  19. (x + 2)(x + 6)
  20. 4 and 6
  21. 625
  22. 63
  23. 4
  24. 16
  25. 1
  26. 4489
  27. 36
  28. (x + 2)(x + 3)
  29. 25
  30. 58
  31. 5329
  32. 1156
  33. (x + 8)(x + 9)
  34. 4
  35. 54
  36. 30
  37. 6084
  38. 10201
  39. 85
  40. 11

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