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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 993 · 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 is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
  2. 2.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
  3. 3.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  4. 4.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
  5. 5.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
  6. 6.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9409 (d) 9991
  7. 7.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  8. 8.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)
  9. 9.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  10. 10.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 100 (d) 4
  11. 11.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
  12. 12.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
  13. 13.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  14. 14.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 193 (c) 361 (d) 277
  15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  16. 16.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
  18. 18.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 121 (c) 93 (d) 177
  19. 19.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 36 (d) 6
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
  21. 21.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 42 (d) 13
  22. 22.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 132 (c) 100 (d) 84
  23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  24. 24.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 133 (c) 122 (d) 144
  25. 25.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
  26. 26.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 97 (c) 65 (d) 81
  27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
  28. 28.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 7921 (d) 8291
  29. 29.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 52 (d) 100
  30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
  31. 31.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 20 (c) 36 (d) 4
  32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  33. 33.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 5 (c) 13 (d) 9
  34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12
  36. 36.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 5 (d) 1 and 8
  37. 37.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4225 (d) 4875
  38. 38.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
  39. 39.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 333 (d) 117
  40. 40.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 81 (c) 225 (d) 9

Answer key

  1. 12
  2. 15
  3. 89
  4. 74
  5. 625
  6. 9409
  7. 1
  8. (x + 7)(x + 9)
  9. 20
  10. 4
  11. 7056
  12. 208
  13. (x + 2)(x + 8)
  14. 193
  15. (x + 4)(x + 8)
  16. (x + 2)(x + 9)
  17. 225
  18. 65
  19. 36
  20. 145
  21. 42
  22. 68
  23. (x + 6)(x + 9)
  24. 122
  25. 6724
  26. 65
  27. 484
  28. 8281
  29. 52
  30. (x + 3)(x + 6)
  31. 4
  32. (x + 5)(x + 7)
  33. 5
  34. (x + 3)(x + 8)
  35. 13
  36. 2 and 4
  37. 4225
  38. (x + 5)(x + 8)
  39. 117
  40. 81

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