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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 268 · 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.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)
  2. 2.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 12 (d) 8
  3. 3.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 3025 (c) 2025 (d) 2475
  5. 5.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
  7. 7.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 6425 (d) 5625
  8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
  9. 9.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 352 (c) 208 (d) 256
  10. 10.If a + b = 13 and ab = 30, what is (a − b)²? (a) 49 (b) 169 (c) 7 (d) 109
  11. 11.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 45 (d) 1 and 13
  12. 12.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
  13. 13.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 130 (d) 256
  14. 14.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 121 (c) 169 (d) 145
  15. 15.If a + b = 9 and ab = 8, what is (a − b)²? (a) 65 (b) 7 (c) 49 (d) 81
  16. 16.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 57 (c) 78 (d) 50
  17. 17.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
  18. 18.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)
  19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  20. 20.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 36 (d) 37
  21. 21.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 81 (c) 73 (d) 97
  22. 22.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 117 (d) 333
  23. 23.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3721 (d) 3731
  24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 16 (c) 15 (d) 63
  25. 25.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 6 (d) 100
  26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  27. 27.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 2 (d) 4
  28. 28.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3249 (c) 3591 (d) 3259
  29. 29.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
  30. 30.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)
  31. 31.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 40 (c) 1 and 12 (d) 2 and 20
  32. 32.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 404 (d) 494
  33. 33.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
  34. 34.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 289 (d) 157
  35. 35.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  36. 36.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  37. 37.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 361 (d) 193
  38. 38.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 33 (d) 32
  39. 39.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 73 (c) 121 (d) 5
  40. 40.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 53 (d) 109

Answer key

  1. (x + 7)(x + 9)
  2. 7
  3. (x + 2)(x + 4)
  4. 2025
  5. (x + 3)(x + 4)
  6. 1936
  7. 7225
  8. 8281
  9. 160
  10. 49
  11. 5 and 9
  12. 8 and 9
  13. 130
  14. 121
  15. 49
  16. 50
  17. 61
  18. (x + 6)(x + 7)
  19. 14
  20. 36
  21. 65
  22. 117
  23. 3721
  24. 16
  25. 36
  26. 145
  27. 4
  28. 3249
  29. 6561
  30. (x + 8)(x + 9)
  31. 5 and 8
  32. 484
  33. 13
  34. 25
  35. 170
  36. (x + 5)(x + 8)
  37. 25
  38. 32
  39. 25
  40. 53

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