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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 803 · 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.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
  2. 2.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 37 (c) 13 (d) 36
  3. 3.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 23 (c) 10 (d) 24
  4. 4.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 851 (c) 899 (d) 961
  5. 5.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7931 (d) 7921
  6. 6.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 37 (d) 49
  7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  8. 8.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)
  9. 9.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 7 (d) 9
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  11. 11.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
  12. 12.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5329 (c) 4909 (d) 5339
  13. 13.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 37 (c) 49 (d) 25
  14. 14.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 169 (c) 49 (d) 109
  15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  16. 16.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 529 (d) 361
  17. 17.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 4096 (d) 3616
  18. 18.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 61 (d) 91
  19. 19.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 8
  20. 20.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2411 (c) 2401 (d) 2499
  21. 21.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2484 (d) 2116
  22. 22.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  23. 23.What is the product of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 17 (d) 15
  24. 24.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
  25. 25.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 5 (d) 1 and 8
  26. 26.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 1 (c) 169 (d) 225
  27. 27.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 625 (d) 1225
  28. 28.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
  29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7569 (d) 7579
  31. 31.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 157 (c) 85 (d) 103
  32. 32.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 2 (d) 16
  33. 33.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
  34. 34.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 1575 (c) 1235 (d) 2025
  35. 35.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 100 (d) 36
  36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
  37. 37.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
  38. 38.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6241 (d) 6399
  39. 39.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 1 and 54 (d) 6 and 9
  40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)

Answer key

  1. 7 and 8
  2. 36
  3. 24
  4. 841
  5. 7921
  6. 37
  7. (x + 6)(x + 7)
  8. (x + 7)(x + 9)
  9. 8
  10. 16
  11. 113
  12. 5329
  13. 1
  14. 49
  15. (x + 6)(x + 8)
  16. 193
  17. 4096
  18. 61
  19. 12
  20. 2401
  21. 2116
  22. (x + 3)(x + 8)
  23. 16
  24. 1156
  25. 2 and 4
  26. 1
  27. 625
  28. 2304
  29. (x + 3)(x + 5)
  30. 7569
  31. 85
  32. 4
  33. 48
  34. 1225
  35. 36
  36. 441
  37. 3364
  38. 6241
  39. 6 and 9
  40. (x + 5)(x + 6)

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