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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 735 · 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 = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 148 (d) 100
  2. 2.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
  3. 3.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
  4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  5. 5.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 90 (c) 6 (d) 36
  6. 6.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 2025 (d) 1575
  7. 7.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 539 (d) 289
  8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1604 (d) 1444
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
  12. 12.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 4 (c) 2 and 3 (d) 1 and 4
  13. 13.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8474 (d) 8464
  14. 14.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
  15. 15.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  16. 16.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2504 (d) 2714
  18. 18.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x + 1)(x + 9)
  19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  20. 20.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
  21. 21.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
  23. 23.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 6 (c) 3 and 4 (d) 1 and 12
  24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  25. 25.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 47 (c) 49 (d) 48
  26. 26.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 10 (c) 144 (d) 100
  27. 27.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4901 (d) 4761
  28. 28.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 18 (d) 12
  29. 29.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 235 (c) 225 (d) 625
  30. 30.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 9 (d) 14
  31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
  32. 32.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 7056 (d) 6416
  34. 34.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 61 (c) 81 (d) 121
  35. 35.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 9
  36. 36.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)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4356 (d) 4366
  38. 38.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
  39. 39.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 4 (d) 144
  40. 40.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

Answer key

  1. 100
  2. 841
  3. 9
  4. (x + 2)(x + 6)
  5. 36
  6. 1225
  7. 529
  8. (x + 5)(x + 6)
  9. 1764
  10. (x + 8)(x + 9)
  11. 2916
  12. 2 and 3
  13. 8464
  14. (x + 3)(x + 6)
  15. 6
  16. 34
  17. 2704
  18. (x + 3)(x + 7)
  19. 16
  20. 225
  21. (x + 4)(x + 6)
  22. 3481
  23. 3 and 4
  24. (x + 7)(x + 9)
  25. 48
  26. 100
  27. 5041
  28. 11
  29. 225
  30. 9
  31. 145
  32. (x + 4)(x + 8)
  33. 7056
  34. 41
  35. 8
  36. (x + 6)(x + 9)
  37. 4356
  38. 16
  39. 4
  40. 8 and 9

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