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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 908 · 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.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 10 (c) 13 (d) 16
  2. 2.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 13 (d) 7
  3. 3.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 56 (d) 14
  4. 4.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
  5. 5.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 2 and 4 (c) 1 and 4 (d) 1 and 6
  6. 6.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)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2484 (d) 2916
  8. 8.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10001 (d) 10201
  9. 9.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
  10. 10.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
  11. 11.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 891 (d) 729
  13. 13.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 25 (d) 17
  14. 14.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
  15. 15.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4634 (c) 4896 (d) 5184
  16. 16.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 7 and 8 (d) 2 and 28
  17. 17.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  18. 18.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2314 (c) 2704 (d) 2496
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  20. 20.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
  21. 21.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 10
  22. 22.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 5 (c) 49 (d) 25
  23. 23.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  24. 24.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1936 (d) 1616
  25. 25.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 100 (c) 76 (d) 148
  26. 26.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
  27. 27.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 8
  28. 28.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 33 (d) 32
  29. 29.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
  30. 30.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
  32. 32.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 153 (d) 297
  33. 33.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
  34. 34.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 144 (d) 90
  35. 35.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 9 (c) 101 (d) 81
  36. 36.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
  38. 38.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 7 (d) 10
  39. 39.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
  40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)

Answer key

  1. 10
  2. 5
  3. 15
  4. 324
  5. 2 and 3
  6. (x + 5)(x + 8)
  7. 2116
  8. 10201
  9. 3 and 4
  10. 5 and 6
  11. (x + 4)(x + 7)
  12. 729
  13. 17
  14. 11
  15. 4624
  16. 7 and 8
  17. 56
  18. 2304
  19. (x + 7)(x + 8)
  20. 8 and 9
  21. 2 and 9
  22. 25
  23. (x + 8)(x + 9)
  24. 1936
  25. 52
  26. 3721
  27. 12
  28. 32
  29. 1444
  30. (x + 4)(x + 9)
  31. 8281
  32. 153
  33. 10609
  34. 90
  35. 81
  36. (x + 3)(x + 4)
  37. 7225
  38. 7
  39. 6
  40. (x + 7)(x + 9)

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