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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 977 · 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 − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 4 (c) 2 and 5 (d) 1 and 8
  2. 2.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 841 (d) 961
  3. 3.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  6. 6.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
  7. 7.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
  9. 9.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 46 (c) 45 (d) 14
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  11. 11.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 27 (c) 11 (d) 28
  12. 12.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 529 (d) 539
  13. 13.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 97 (d) 241
  14. 14.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 7 (d) 5
  15. 15.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 85 (c) 121 (d) 49
  16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  17. 17.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 117 (c) 63 (d) 45
  18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  19. 19.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 36 (c) 50 (d) 6
  20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
  21. 21.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  23. 23.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 52 (c) 64 (d) 40
  24. 24.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  25. 25.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)
  26. 26.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
  27. 27.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 85 (d) 169
  28. 28.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 11
  29. 29.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)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
  31. 31.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 657 (d) 441
  32. 32.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 16 (d) 11
  33. 33.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 4 (c) 16 (d) 136
  34. 34.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
  35. 35.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81
  36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6241 (c) 6561 (d) 6251
  38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 57 (d) 55
  39. 39.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
  40. 40.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 6 and 9 (d) 1 and 54

Answer key

  1. 2 and 4
  2. 841
  3. (x + 2)(x + 4)
  4. 17
  5. 16
  6. 30
  7. (x + 2)(x + 9)
  8. 15
  9. 45
  10. (x + 8)(x + 9)
  11. 28
  12. 529
  13. 97
  14. 5
  15. 49
  16. (x + 4)(x + 5)
  17. 45
  18. (x + 5)(x + 6)
  19. 36
  20. 1089
  21. 9
  22. (x + 7)(x + 9)
  23. 40
  24. 14
  25. (x + 4)(x + 6)
  26. 7 and 8
  27. 1
  28. 24
  29. (x + 5)(x + 8)
  30. 5184
  31. 225
  32. 10
  33. 16
  34. 116
  35. 9
  36. (x + 7)(x + 8)
  37. 6241
  38. 56
  39. 136
  40. 6 and 9

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