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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 35 · 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 27²? (a) 739 (b) 1089 (c) 891 (d) 729
  2. 2.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 6375 (c) 7225 (d) 5635
  3. 3.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  5. 5.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  6. 6.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
  7. 7.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 235 (c) 375 (d) 625
  8. 8.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 18 (c) 19 (d) 11
  9. 9.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  10. 10.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)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2209 (c) 2819 (d) 2809
  12. 12.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  13. 13.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 84 (c) 132 (d) 68
  14. 14.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 8 (d) 7
  15. 15.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 409 (c) 169 (d) 289
  16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  17. 17.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 19 (d) 25
  18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  19. 19.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
  20. 20.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 7 (c) 169 (d) 109
  21. 21.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 382 (d) 130
  22. 22.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 4 (d) 34
  23. 23.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  24. 24.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 208 (c) 304 (d) 400
  25. 25.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3969 (d) 3249
  26. 26.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 1225 (d) 625
  27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  28. 28.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 169 (c) 113 (d) 225
  29. 29.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2304 (c) 2704 (d) 2314
  30. 30.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 6391 (d) 5939
  31. 31.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 144 (c) 100 (d) 10
  32. 32.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 1 and 56 (d) 2 and 28
  33. 33.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 4106 (c) 3136 (d) 3616
  34. 34.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10004 (d) 10404
  35. 35.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
  36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  37. 37.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 1 (d) 19
  38. 38.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  39. 39.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 1 and 12 (d) 6 and 7
  40. 40.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 37 (c) 49 (d) 43

Answer key

  1. 729
  2. 5625
  3. 3721
  4. (x + 8)(x + 9)
  5. (x + 3)(x + 8)
  6. 1
  7. 225
  8. 18
  9. (x + 2)(x + 4)
  10. (x + 4)(x + 8)
  11. 2809
  12. 9
  13. 68
  14. 7
  15. 169
  16. (x + 4)(x + 6)
  17. 13
  18. (x + 4)(x + 5)
  19. (x + 5)(x + 8)
  20. 49
  21. 130
  22. 4
  23. (x + 5)(x + 9)
  24. 208
  25. 3249
  26. 625
  27. (x + 7)(x + 9)
  28. 113
  29. 2304
  30. 5929
  31. 100
  32. 7 and 8
  33. 4096
  34. 10404
  35. 1156
  36. (x + 6)(x + 8)
  37. 1
  38. 1
  39. 6 and 7
  40. 37

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