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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 715 · 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 19²? (a) 399 (b) 441 (c) 371 (d) 361
  2. 2.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 3 and 6 (d) 2 and 9
  3. 3.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)
  4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  7. 7.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
  8. 8.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 196 (c) 130 (d) 64
  9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  10. 10.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 13 (d) 40
  11. 11.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
  12. 12.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)
  13. 13.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 68 (c) 100 (d) 6
  14. 14.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 4901 (c) 5041 (d) 5051
  15. 15.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  16. 16.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 185 (d) 9
  17. 17.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  18. 18.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 36 (d) 6
  19. 19.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 81 (c) 7 (d) 49
  20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  21. 21.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  22. 22.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 157 (c) 85 (d) 121
  23. 23.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  24. 24.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)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 225 (c) 235 (d) 375
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  27. 27.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  28. 28.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
  29. 29.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 177 (d) 121
  30. 30.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
  31. 31.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 247 (d) 170
  32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 598 (d) 301
  33. 33.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 7225 (c) 7235 (d) 9025
  34. 34.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 4 (d) 208
  35. 35.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
  36. 36.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  37. 37.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 391 (d) 289
  38. 38.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 225 (c) 153 (d) 81
  39. 39.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 15 (d) 3 and 5
  40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271

Answer key

  1. 361
  2. 2 and 9
  3. (x + 3)(x + 8)
  4. 63
  5. 15
  6. (x + 8)(x + 9)
  7. 4
  8. 64
  9. 56
  10. 13
  11. 225
  12. (x + 3)(x + 6)
  13. 36
  14. 5041
  15. (x + 2)(x + 6)
  16. 9
  17. 1
  18. 36
  19. 49
  20. (x + 3)(x + 7)
  21. (x + 4)(x + 8)
  22. 85
  23. (x + 6)(x + 8)
  24. (x + 7)(x + 8)
  25. 225
  26. (x + 2)(x + 7)
  27. 164
  28. 145
  29. 65
  30. 16
  31. 170
  32. 202
  33. 7225
  34. 16
  35. 484
  36. 1089
  37. 289
  38. 81
  39. 3 and 5
  40. 1

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