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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 234 · 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 = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 85 (d) 127
  2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
  3. 3.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 97 (d) 65
  4. 4.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 31 (d) 11
  5. 5.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
  6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 57 (d) 55
  7. 7.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
  8. 8.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8659 (d) 8109
  9. 9.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 17 (c) 25 (d) 21
  10. 10.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 81 (d) 1
  11. 11.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3249 (c) 3591 (d) 3969
  12. 12.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 841 (d) 971
  13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 416 (d) 256
  15. 15.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 26 (d) 28
  16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3364 (c) 3604 (d) 3854
  18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  22. 22.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 44 (d) 46
  23. 23.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
  24. 24.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 8 (d) 9
  25. 25.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2501 (d) 2601
  26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
  27. 27.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
  28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  29. 29.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 409 (d) 539
  31. 31.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
  32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
  33. 33.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 961 (d) 851
  34. 34.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 185 (d) 3
  35. 35.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 9025 (d) 11025
  36. 36.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 144 (d) 112
  37. 37.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 12 (d) 1 and 7
  38. 38.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
  39. 39.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 91 (d) 61
  40. 40.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)

Answer key

  1. 1
  2. 15
  3. 65
  4. 30
  5. 361
  6. 56
  7. 26
  8. 8649
  9. 17
  10. 1
  11. 3249
  12. 961
  13. (x + 3)(x + 4)
  14. 576
  15. 27
  16. (x + 4)(x + 6)
  17. 3844
  18. 1
  19. (x + 4)(x + 8)
  20. (x + 8)(x + 9)
  21. 676
  22. 45
  23. 5 and 7
  24. 8
  25. 2601
  26. (x + 7)(x + 8)
  27. 225
  28. (x + 7)(x + 9)
  29. (x + 3)(x + 8)
  30. 529
  31. (x + 2)(x + 9)
  32. 202
  33. 841
  34. 9
  35. 9025
  36. 80
  37. 2 and 6
  38. 65
  39. 61
  40. (x + 3)(x + 9)

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