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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 265 · 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.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 4235 (d) 3025
  2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  3. 3.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 11 (c) 25 (d) 24
  4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  5. 5.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 2 (d) 100
  6. 6.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 6 (c) 1 and 12 (d) 3 and 4
  7. 7.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 164 (d) 2
  8. 8.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
  9. 9.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)
  10. 10.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 400 (d) 208
  11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 794 (d) 896
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  14. 14.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 8 (d) 6
  15. 15.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
  16. 16.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
  17. 17.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 106 (c) 196 (d) 151
  18. 18.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
  19. 19.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
  20. 20.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 8 (d) 12
  21. 21.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 676 (d) 1156
  22. 22.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  23. 23.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 12 (d) 1 and 40
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  25. 25.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  26. 26.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 391 (d) 289
  27. 27.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 144 (d) 74
  28. 28.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 7931 (d) 8099
  29. 29.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 12 (c) 1 and 42 (d) 2 and 21
  30. 30.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
  31. 31.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  32. 32.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 16 (c) 64 (d) 4
  33. 33.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 256 (c) 266 (d) 384
  34. 34.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 2025 (d) 1575
  35. 35.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 235 (d) 225
  36. 36.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
  37. 37.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  38. 38.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 33 (c) 32 (d) 12
  39. 39.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 170 (d) 247
  40. 40.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10826 (d) 10016

Answer key

  1. 4225
  2. 72
  3. 24
  4. (x + 4)(x + 6)
  5. 4
  6. 3 and 4
  7. 4
  8. 5 and 6
  9. (x + 3)(x + 8)
  10. 208
  11. (x + 2)(x + 4)
  12. 784
  13. (x + 8)(x + 9)
  14. 6
  15. 15
  16. 11
  17. 106
  18. 63
  19. 42
  20. 8
  21. 1156
  22. (x + 5)(x + 7)
  23. 5 and 8
  24. 17
  25. 6
  26. 289
  27. 74
  28. 7921
  29. 6 and 7
  30. 361
  31. 56
  32. 16
  33. 256
  34. 1225
  35. 225
  36. 441
  37. (x + 2)(x + 3)
  38. 32
  39. 170
  40. 10816

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