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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 293 · 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 product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
  3. 3.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 4225 (d) 3575
  4. 4.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 37 (d) 43
  5. 5.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
  6. 6.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
  7. 7.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 36 (d) 35
  8. 8.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 8 (d) 7
  9. 9.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 4 (d) 16
  10. 10.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 256 (d) 4
  11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  12. 12.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 81 (d) 9
  13. 13.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 15 (d) 54
  14. 14.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 73 (d) 25
  15. 15.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 635 (d) 625
  16. 16.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 181 (d) 91
  17. 17.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
  18. 18.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 598 (d) 202
  19. 19.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
  20. 20.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  23. 23.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 262 (c) 130 (d) 163
  24. 24.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 125 (c) 225 (d) 25
  25. 25.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 625 (c) 375 (d) 235
  26. 26.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 225 (c) 3 (d) 9
  27. 27.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1604 (c) 1764 (d) 1774
  28. 28.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2491 (c) 2809 (d) 2209
  29. 29.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2411 (d) 2601
  30. 30.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 4 (c) 16 (d) 106
  31. 31.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 9 (c) 7 (d) 5
  32. 32.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 4225 (d) 3025
  33. 33.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  34. 34.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
  35. 35.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 7406 (c) 8084 (d) 8836
  36. 36.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
  37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
  38. 38.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
  39. 39.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
  40. 40.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 2 and 36 (d) 8 and 9

Answer key

  1. 56
  2. 72
  3. 3025
  4. 37
  5. 13
  6. 113
  7. 36
  8. 8
  9. 16
  10. 16
  11. (x + 5)(x + 8)
  12. 81
  13. 15
  14. 25
  15. 625
  16. 61
  17. 16
  18. 202
  19. 42
  20. (x + 5)(x + 6)
  21. (x + 2)(x + 3)
  22. (x + 7)(x + 9)
  23. 130
  24. 25
  25. 225
  26. 9
  27. 1764
  28. 2209
  29. 2401
  30. 16
  31. 1
  32. 4225
  33. 6
  34. 14
  35. 7396
  36. (x + 4)(x + 6)
  37. 63
  38. 361
  39. 4
  40. 8 and 9

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