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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 807 · 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 = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 2 (c) 100 (d) 4
  2. 2.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 5 (d) 7
  3. 3.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
  4. 4.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 1 and 12 (d) 2 and 18
  5. 5.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  7. 7.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2411 (d) 2499
  9. 9.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8464 (d) 8474
  10. 10.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  11. 11.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 4 (d) 58
  12. 12.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 324 (c) 164 (d) 244
  13. 13.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
  15. 15.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)
  16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
  17. 17.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 14 (d) 15
  18. 18.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 9025 (c) 7235 (d) 8075
  19. 19.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 2 and 8 (c) 3 and 7 (d) 1 and 16
  20. 20.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 149 (d) 9
  21. 21.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 144 (d) 184
  22. 22.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
  23. 23.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  24. 24.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 1 and 9 (d) 3 and 7
  25. 25.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  27. 27.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 4 (c) 1 and 5 (d) 2 and 5
  28. 28.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 6
  29. 29.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  30. 30.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 142 (c) 58 (d) 79
  31. 31.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 384 (c) 266 (d) 576
  33. 33.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 27 (d) 12
  34. 34.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10404 (c) 9604 (d) 10004
  35. 35.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
  36. 36.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 73 (c) 25 (d) 5
  37. 37.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2611 (d) 2501
  38. 38.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
  39. 39.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 7056 (d) 5786
  40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164

Answer key

  1. 4
  2. 5
  3. 8 and 9
  4. 4 and 9
  5. 1
  6. 56
  7. (x + 5)(x + 6)
  8. 2401
  9. 8464
  10. (x + 2)(x + 8)
  11. 16
  12. 164
  13. (x + 4)(x + 8)
  14. (x + 5)(x + 9)
  15. (x + 8)(x + 9)
  16. (x + 4)(x + 9)
  17. 15
  18. 7225
  19. 2 and 8
  20. 9
  21. 104
  22. 40
  23. 12
  24. 3 and 7
  25. (x + 3)(x + 8)
  26. (x + 7)(x + 9)
  27. 2 and 4
  28. 7
  29. 576
  30. 58
  31. (x + 2)(x + 3)
  32. 256
  33. 12
  34. 10404
  35. 13
  36. 25
  37. 2601
  38. 28
  39. 5776
  40. 4

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