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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 83 · 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.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 169 (d) 129
  2. 2.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 241 (c) 133 (d) 169
  3. 3.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 36 (c) 52 (d) 28
  4. 4.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2916 (d) 2926
  5. 5.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3364 (d) 3596
  6. 6.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
  7. 7.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
  8. 8.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 117 (c) 81 (d) 63
  9. 9.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 5929 (c) 6391 (d) 6889
  10. 10.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 164 (d) 324
  11. 11.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 421 (d) 157
  12. 12.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 58 (c) 4 (d) 16
  13. 13.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3844 (d) 3854
  14. 14.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 100 (c) 6 (d) 36
  15. 15.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 73 (c) 169 (d) 121
  16. 16.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
  17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  18. 18.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 494 (d) 324
  19. 19.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2525 (d) 2025
  20. 20.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  22. 22.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4489 (c) 4499 (d) 4891
  24. 24.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8291 (c) 8281 (d) 8101
  25. 25.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 3 (c) 117 (d) 225
  26. 26.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
  27. 27.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 4 and 6 (d) 1 and 9
  28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  30. 30.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 289 (d) 433
  32. 32.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
  33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  34. 34.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4096 (d) 4106
  36. 36.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  37. 37.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 13 (d) 45
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 441
  39. 39.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 14)
  40. 40.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)

Answer key

  1. 89
  2. 97
  3. 20
  4. 2916
  5. 3364
  6. 5 and 7
  7. 5041
  8. 45
  9. 5929
  10. 164
  11. 157
  12. 16
  13. 3844
  14. 36
  15. 73
  16. 7 and 8
  17. 72
  18. 484
  19. 3025
  20. (x + 4)(x + 9)
  21. (x + 8)(x + 9)
  22. (x + 2)(x + 5)
  23. 4489
  24. 8281
  25. 9
  26. 5
  27. 4 and 6
  28. (x + 6)(x + 9)
  29. (x + 6)(x + 8)
  30. 181
  31. 145
  32. 9409
  33. 17
  34. (x + 5)(x + 8)
  35. 4096
  36. 36
  37. 14
  38. 9
  39. (x + 2)(x + 7)
  40. (x + 3)(x + 5)

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