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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 896 · 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 − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
  2. 2.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8649 (d) 8659
  3. 3.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 484 (d) 324
  5. 5.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3025 (d) 3575
  6. 6.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 1 (d) 121
  7. 7.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 117 (d) 225
  8. 8.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 4 (c) 36 (d) 16
  9. 9.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 14 (d) 13
  10. 10.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6724 (c) 6084 (d) 6734
  11. 11.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 81 (d) 3
  12. 12.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 18 (c) 11 (d) 19
  13. 13.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 144 (c) 100 (d) 10
  14. 14.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 80 (d) 208
  15. 15.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 106 (c) 151 (d) 286
  16. 16.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  19. 19.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 6 (d) 5
  20. 20.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 4 (c) 106 (d) 196
  21. 21.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 15 (d) 1 and 7
  22. 22.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 4 (d) 5
  23. 23.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9999 (d) 9801
  24. 24.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 193 (d) 5
  25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
  26. 26.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 49 (c) 73 (d) 25
  27. 27.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 46 (d) 36
  28. 28.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 181 (d) 137
  29. 29.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 144 (d) 109
  30. 30.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 225 (c) 113 (d) 169
  31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
  32. 32.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 592 (d) 400
  33. 33.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
  34. 34.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 58 (d) 100
  35. 35.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 235 (d) 375
  36. 36.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
  38. 38.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 201 (c) 146 (d) 256
  39. 39.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 10 (d) 20
  40. 40.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 198 (d) 90

Answer key

  1. 7 and 9
  2. 8649
  3. (x + 3)(x + 4)
  4. 164
  5. 3025
  6. 1
  7. 117
  8. 16
  9. 14
  10. 6724
  11. 9
  12. 18
  13. 100
  14. 80
  15. 106
  16. (x + 5)(x + 9)
  17. (x + 8)(x + 9)
  18. (x + 7)(x + 8)
  19. 6
  20. 16
  21. 3 and 5
  22. 5
  23. 9801
  24. 25
  25. (x + 3)(x + 5)
  26. 25
  27. 26
  28. 137
  29. 74
  30. 1
  31. (x + 2)(x + 7)
  32. 208
  33. 202
  34. 58
  35. 225
  36. (x + 2)(x + 4)
  37. 63
  38. 146
  39. 9
  40. 90

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