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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 181 · 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.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4106 (c) 4096 (d) 3136
  3. 3.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
  4. 4.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 169 (d) 85
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
  6. 6.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 9 (d) 8
  7. 7.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  8. 8.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 7396 (c) 8846 (d) 8116
  10. 10.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 6
  11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3979 (d) 3609
  12. 12.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
  13. 13.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
  14. 14.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
  15. 15.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
  16. 16.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 40 (c) 4 (d) 64
  17. 17.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 73 (d) 25
  18. 18.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 10
  19. 19.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 4916 (d) 5476
  20. 20.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 1 and 6 (d) 2 and 3
  21. 21.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 625 (d) 925
  22. 22.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721
  23. 23.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
  24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  26. 26.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 41 (d) 42
  27. 27.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 6375 (c) 7225 (d) 5625
  28. 28.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180
  29. 29.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
  30. 30.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 58 (d) 142
  31. 31.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 1166 (d) 916
  32. 32.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 3 and 6 (d) 2 and 7
  33. 33.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  34. 34.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
  35. 35.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
  36. 36.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 64 (d) 50
  37. 37.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 11 (c) 145 (d) 121
  38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3025 (c) 4225 (d) 3035
  39. 39.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 11 (c) 9 (d) 16
  40. 40.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 196 (d) 172

Answer key

  1. (x + 6)(x + 7)
  2. 4096
  3. 4
  4. 1
  5. 15
  6. 8
  7. (x + 4)(x + 8)
  8. (x + 3)(x + 9)
  9. 8836
  10. 3 and 4
  11. 3969
  12. (x + 4)(x + 9)
  13. 4
  14. 16
  15. 17
  16. 16
  17. 25
  18. 24
  19. 5476
  20. 2 and 3
  21. 1225
  22. 3721
  23. 12
  24. (x + 8)(x + 9)
  25. (x + 6)(x + 9)
  26. 42
  27. 5625
  28. 36
  29. (x + 3)(x + 5)
  30. 58
  31. 1156
  32. 2 and 7
  33. (x + 3)(x + 8)
  34. 20
  35. 145
  36. 50
  37. 121
  38. 3025
  39. 10
  40. 148

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