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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 195 · 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 + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  5. 5.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 8281 (c) 7931 (d) 7921
  6. 6.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 8 (c) 6 (d) 7
  7. 7.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
  8. 8.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 14 (d) 49
  9. 9.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 2 and 5 (c) 1 and 6 (d) 1 and 10
  10. 10.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)
  11. 11.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  12. 12.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
  13. 13.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 91 (c) 100 (d) 118
  14. 14.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 16 (c) 26 (d) 4
  15. 15.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 252 (d) 468
  16. 16.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  17. 17.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 64 (c) 16 (d) 40
  18. 18.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 289 (c) 223 (d) 157
  19. 19.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 9 (c) 10 (d) 11
  20. 20.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 884 (c) 1156 (d) 686
  21. 21.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
  22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2611 (c) 2601 (d) 2501
  23. 23.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 4 (d) 400
  24. 24.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
  25. 25.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 301 (d) 598
  26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
  27. 27.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 262 (d) 163
  28. 28.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 539 (d) 289
  29. 29.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 100 (d) 58
  30. 30.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
  31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  32. 32.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 361 (d) 273
  33. 33.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 9216 (d) 10816
  34. 34.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 27 (d) 13
  35. 35.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 7 (d) 8
  36. 36.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3844 (d) 3604
  37. 37.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5625 (c) 5635 (d) 4225
  38. 38.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 794 (d) 896
  39. 39.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  40. 40.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 11 (c) 169 (d) 145

Answer key

  1. (x + 4)(x + 6)
  2. (x + 7)(x + 8)
  3. (x + 5)(x + 9)
  4. 56
  5. 7921
  6. 8
  7. 164
  8. 48
  9. 2 and 5
  10. (x + 7)(x + 9)
  11. (x + 4)(x + 5)
  12. 145
  13. 82
  14. 16
  15. 180
  16. (x + 3)(x + 8)
  17. 16
  18. 157
  19. 10
  20. 676
  21. 14
  22. 2601
  23. 16
  24. (x + 5)(x + 7)
  25. 202
  26. 7
  27. 130
  28. 529
  29. 58
  30. 49
  31. (x + 5)(x + 6)
  32. 185
  33. 10816
  34. 12
  35. 8
  36. 3844
  37. 5625
  38. 784
  39. (x + 8)(x + 9)
  40. 121

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