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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 628 · 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 − 17x + 72 = 0? (a) 17 (b) 72 (c) 73 (d) 71
  2. 2.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
  3. 3.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 85 (d) 253
  4. 4.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
  5. 5.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 193 (d) 256
  6. 6.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 82 (c) 64 (d) 8
  7. 7.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 2 (d) 144
  8. 8.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
  9. 9.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 441 (d) 225
  10. 10.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 49 (c) 37 (d) 25
  11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  12. 12.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 16 (c) 256 (d) 136
  13. 13.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 4 (d) 16
  14. 14.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 2525 (d) 3035
  15. 15.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 49 (c) 7 (d) 121
  16. 16.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 81 (d) 41
  17. 17.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 18 (d) 10
  18. 18.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 196 (c) 172 (d) 244
  19. 19.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  21. 21.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 29 (c) 28 (d) 27
  22. 22.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2704 (d) 2714
  24. 24.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 7 and 8 (d) 1 and 14
  25. 25.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  26. 26.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 97 (c) 25 (d) 169
  27. 27.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19
  28. 28.If a + b = 15 and ab = 36, what is a² + b²? (a) 153 (b) 225 (c) 297 (d) 189
  29. 29.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 6384 (d) 5786
  30. 30.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  31. 31.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  32. 32.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)
  33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  34. 34.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 8075 (d) 7225
  35. 35.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 121 (c) 3 (d) 9
  36. 36.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  38. 38.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 5625 (d) 6425
  39. 39.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1616 (d) 1296
  40. 40.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11

Answer key

  1. 72
  2. 202
  3. 85
  4. 9
  5. 130
  6. 64
  7. 4
  8. 3969
  9. 225
  10. 1
  11. (x + 3)(x + 5)
  12. 16
  13. 4
  14. 3025
  15. 49
  16. 1
  17. 11
  18. 148
  19. 289
  20. (x + 8)(x + 9)
  21. 28
  22. (x + 4)(x + 9)
  23. 2704
  24. 7 and 8
  25. (x + 3)(x + 8)
  26. 25
  27. 18
  28. 153
  29. 5776
  30. (x + 6)(x + 9)
  31. (x + 2)(x + 8)
  32. (x + 7)(x + 9)
  33. (x + 4)(x + 5)
  34. 7225
  35. 9
  36. 10
  37. 181
  38. 7225
  39. 1936
  40. 11

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