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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 79 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  2. 2.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
  3. 3.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 85 (c) 157 (d) 103
  4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  5. 5.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  7. 7.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 148 (c) 196 (d) 10
  8. 8.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521
  9. 9.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 64 (c) 40 (d) 16
  10. 10.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 20 (d) 19
  11. 11.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 35 (d) 12
  12. 12.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 160 (c) 256 (d) 208
  13. 13.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
  14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  15. 15.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
  16. 16.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 169 (d) 85
  17. 17.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 441 (d) 9
  18. 18.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 9 (c) 15 (d) 14
  19. 19.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
  20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  21. 21.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 13 (c) 3 and 15 (d) 1 and 45
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  23. 23.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  24. 24.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 46 (d) 14
  25. 25.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 64 (d) 50
  26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7066 (c) 7056 (d) 5776
  27. 27.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 8 (d) 6
  28. 28.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 1156 (d) 884
  30. 30.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)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
  34. 34.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 361 (d) 25
  35. 35.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 20 (c) 36 (d) 52
  36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 478 (d) 170
  37. 37.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 27 (d) 28
  38. 38.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 851 (d) 899
  39. 39.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 2 and 5 (d) 1 and 5
  40. 40.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 157 (d) 223

Answer key

  1. (x + 8)(x + 9)
  2. 12
  3. 85
  4. (x + 7)(x + 8)
  5. 14
  6. (x + 2)(x + 4)
  7. 100
  8. 1521
  9. 16
  10. 20
  11. 35
  12. 160
  13. 3721
  14. 17
  15. 225
  16. 1
  17. 9
  18. 14
  19. 1
  20. 72
  21. 5 and 9
  22. (x + 6)(x + 7)
  23. (x + 4)(x + 7)
  24. 45
  25. 50
  26. 7056
  27. 7
  28. (x + 4)(x + 9)
  29. 676
  30. (x + 6)(x + 9)
  31. 441
  32. (x + 4)(x + 6)
  33. 3025
  34. 25
  35. 20
  36. 170
  37. 27
  38. 841
  39. 2 and 4
  40. 157

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