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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 75 · 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 = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
  2. 2.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1225 (c) 1235 (d) 625
  3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  4. 4.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 6 (d) 9
  5. 5.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 35 (d) 36
  6. 6.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)
  7. 7.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  8. 8.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 5 (d) 157
  9. 9.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)
  10. 10.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 21 (d) 11
  11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  13. 13.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 40 (d) 1 and 12
  14. 14.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 142 (d) 100
  15. 15.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 1 (d) 225
  16. 16.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 676 (d) 1156
  17. 17.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 409 (d) 529
  18. 18.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 4909 (c) 5339 (d) 4489
  19. 19.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1936 (c) 1306 (d) 1584
  20. 20.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
  21. 21.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 9 (d) 12
  22. 22.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 5194 (d) 4624
  23. 23.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 2 and 12 (d) 1 and 24
  24. 24.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 289 (d) 433
  25. 25.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 16 (d) 63
  26. 26.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
  28. 28.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 7225 (d) 6375
  30. 30.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3249 (c) 3259 (d) 3591
  31. 31.What are the roots of x² − 10x + 16 = 0? (a) 2 and 8 (b) 1 and 9 (c) 3 and 7 (d) 1 and 16
  32. 32.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 4 (c) 64 (d) 16
  33. 33.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 18 (d) 12
  34. 34.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  35. 35.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 6 (d) 36
  36. 36.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 484 (c) 494 (d) 324
  37. 37.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 4 and 5 (c) 2 and 10 (d) 1 and 8
  38. 38.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6399 (d) 6241
  39. 39.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
  40. 40.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 61 (d) 81

Answer key

  1. 13
  2. 1225
  3. (x + 4)(x + 5)
  4. 8
  5. 35
  6. (x + 7)(x + 8)
  7. (x + 6)(x + 9)
  8. 25
  9. (x + 2)(x + 6)
  10. 10
  11. (x + 3)(x + 5)
  12. 17
  13. 5 and 8
  14. 58
  15. 1
  16. 1156
  17. 529
  18. 5329
  19. 1296
  20. 7 and 9
  21. 8
  22. 5184
  23. 4 and 6
  24. 145
  25. 63
  26. (x + 7)(x + 9)
  27. 181
  28. (x + 4)(x + 7)
  29. 5625
  30. 3249
  31. 2 and 8
  32. 16
  33. 11
  34. (x + 6)(x + 8)
  35. 36
  36. 484
  37. 4 and 5
  38. 6241
  39. 1
  40. 41

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