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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 221 · 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 = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
  2. 2.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 13 (c) 9 (d) 14
  3. 3.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
  4. 4.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)
  5. 5.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 2 and 10 (c) 1 and 27 (d) 1 and 11
  6. 6.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 41 (d) 43
  7. 7.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 324 (c) 180 (d) 36
  8. 8.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 223 (d) 157
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  11. 11.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  12. 12.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 3 (d) 185
  13. 13.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 2 and 6 (d) 1 and 15
  14. 14.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  15. 15.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  16. 16.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1531 (d) 1681
  17. 17.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 106 (c) 196 (d) 286
  18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  19. 19.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 36 (d) 28
  20. 20.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 169 (d) 89
  21. 21.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 29 (d) 9
  22. 22.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 11 (d) 7
  23. 23.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 325 (d) 225
  24. 24.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 54 (d) 53
  25. 25.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  26. 26.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
  27. 27.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 10 (d) 24
  28. 28.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 25 (d) 193
  29. 29.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)
  30. 30.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 68 (d) 36
  31. 31.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 81 (d) 41
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 5929 (d) 6391
  34. 34.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 9 (c) 19 (d) 20
  35. 35.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 10025 (d) 11035
  36. 36.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 19 (d) 1
  37. 37.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1584 (d) 1296
  38. 38.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  39. 39.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6404 (d) 6724
  40. 40.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 14 (d) 48

Answer key

  1. 164
  2. 14
  3. 63
  4. (x + 5)(x + 8)
  5. 3 and 9
  6. 42
  7. 36
  8. 157
  9. (x + 8)(x + 9)
  10. 576
  11. (x + 4)(x + 5)
  12. 9
  13. 3 and 5
  14. 16
  15. 181
  16. 1521
  17. 106
  18. (x + 6)(x + 8)
  19. 20
  20. 89
  21. 9
  22. 12
  23. 125
  24. 54
  25. 6
  26. 5 and 9
  27. 10
  28. 25
  29. (x + 7)(x + 8)
  30. 36
  31. 1
  32. (x + 5)(x + 9)
  33. 5929
  34. 20
  35. 11025
  36. 1
  37. 1296
  38. 1681
  39. 6724
  40. 14

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