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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 960 · 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 + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10201 (d) 10211
  3. 3.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 225 (d) 125
  4. 4.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
  5. 5.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
  6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  7. 7.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 9 (d) 10
  8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  9. 9.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 113 (d) 169
  10. 10.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  11. 11.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
  13. 13.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 3 and 5 (d) 1 and 7
  14. 14.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
  15. 15.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 9 (c) 18 (d) 19
  16. 16.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 7225 (d) 9035
  17. 17.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
  18. 18.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 7 (d) 1 and 14
  19. 19.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 40 (d) 39
  20. 20.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  21. 21.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 196 (c) 100 (d) 292
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  23. 23.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
  24. 24.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
  25. 25.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 4 and 7 (d) 2 and 14
  26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
  27. 27.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
  28. 28.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235
  29. 29.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 12 (d) 11
  30. 30.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 4225 (d) 3035
  31. 31.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 324 (d) 484
  32. 32.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4489 (c) 4499 (d) 5329
  33. 33.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3721 (d) 3731
  34. 34.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 27 (c) 3 and 9 (d) 1 and 11
  35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  36. 36.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 9 (d) 20
  37. 37.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 81 (c) 63 (d) 45
  38. 38.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 27 (c) 28 (d) 29
  39. 39.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 901 (d) 971
  40. 40.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 9

Answer key

  1. (x + 7)(x + 8)
  2. 10201
  3. 25
  4. 6
  5. 1
  6. (x + 7)(x + 9)
  7. 9
  8. (x + 5)(x + 8)
  9. 1
  10. 1681
  11. (x + 2)(x + 5)
  12. 2916
  13. 3 and 5
  14. 1089
  15. 18
  16. 9025
  17. 48
  18. 2 and 7
  19. 40
  20. (x + 3)(x + 4)
  21. 100
  22. (x + 6)(x + 7)
  23. 2 and 8
  24. 5184
  25. 4 and 7
  26. 193
  27. 36
  28. 1225
  29. 11
  30. 3025
  31. 164
  32. 4489
  33. 3721
  34. 3 and 9
  35. (x + 2)(x + 8)
  36. 20
  37. 45
  38. 28
  39. 961
  40. 10

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