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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 801 · 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 − 11x + 28 = 0? (a) 11 (b) 28 (c) 10 (d) 12
  2. 2.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 41 (d) 43
  3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  4. 4.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  5. 5.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3481 (c) 3721 (d) 3491
  7. 7.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601
  8. 8.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 100 (c) 122 (d) 144
  9. 9.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 8 (d) 64
  10. 10.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 625 (d) 225
  11. 11.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 4 and 6 (c) 1 and 24 (d) 1 and 9
  12. 12.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 34 (d) 36
  13. 13.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  14. 14.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 7 (d) 5
  15. 15.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3854 (d) 3364
  16. 16.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 113 (c) 169 (d) 337
  17. 17.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 4 and 7 (c) 2 and 14 (d) 1 and 28
  18. 18.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 109 (d) 144
  19. 19.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 89 (c) 249 (d) 129
  20. 20.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
  21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  22. 22.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
  23. 23.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 2 and 4 (c) 1 and 6 (d) 1 and 4
  24. 24.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
  25. 25.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 8 (d) 64
  26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  27. 27.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
  28. 28.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 97 (d) 65
  29. 29.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 6889 (d) 5929
  31. 31.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
  32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 400 (d) 301
  33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 657 (d) 333
  34. 34.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 7 and 9 (d) 1 and 15
  35. 35.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2304 (c) 2496 (d) 2704
  36. 36.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 4106 (c) 3136 (d) 3616
  38. 38.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 53 (d) 67
  39. 39.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 4096 (c) 3146 (d) 3584
  40. 40.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 14 (d) 54

Answer key

  1. 11
  2. 42
  3. (x + 5)(x + 8)
  4. 15
  5. (x + 5)(x + 6)
  6. 3481
  7. 3721
  8. 100
  9. 64
  10. 225
  11. 4 and 6
  12. 35
  13. (x + 3)(x + 9)
  14. 5
  15. 3844
  16. 113
  17. 4 and 7
  18. 74
  19. 89
  20. 1156
  21. (x + 2)(x + 6)
  22. 12
  23. 2 and 3
  24. 4
  25. 64
  26. (x + 6)(x + 7)
  27. 8 and 9
  28. 65
  29. (x + 5)(x + 9)
  30. 5929
  31. (x + 4)(x + 6)
  32. 202
  33. 225
  34. 7 and 9
  35. 2304
  36. (x + 5)(x + 7)
  37. 4096
  38. 53
  39. 3136
  40. 15

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