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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 604 · 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 − 13x + 40 = 0? (a) 13 (b) 14 (c) 40 (d) 12
  2. 2.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
  3. 3.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 16 (c) 64 (d) 4
  4. 4.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 109 (d) 169
  5. 5.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 5
  6. 6.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  7. 7.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 10609 (d) 9409
  8. 8.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10016 (d) 10816
  9. 9.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5776 (c) 5786 (d) 6384
  10. 10.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 148 (c) 244 (d) 196
  11. 11.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
  12. 12.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  13. 13.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 36 (d) 6
  14. 14.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
  15. 15.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)
  16. 16.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)
  17. 17.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 90 (d) 144
  18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  19. 19.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 289 (c) 169 (d) 49
  20. 20.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3721 (d) 3481
  21. 21.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4356 (d) 4366
  22. 22.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 10 (c) 11 (d) 9
  23. 23.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
  25. 25.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
  26. 26.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  28. 28.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)
  29. 29.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
  30. 30.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
  31. 31.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 225 (d) 635
  32. 32.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 375 (d) 225
  33. 33.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  34. 34.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8649 (c) 8659 (d) 8109
  35. 35.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 9025 (d) 8075
  36. 36.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 324 (d) 484
  37. 37.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 1 and 40 (d) 5 and 8
  38. 38.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 2 (d) 100
  39. 39.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 13 (c) 6 and 8 (d) 1 and 48
  40. 40.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)

Answer key

  1. 13
  2. 784
  3. 16
  4. 109
  5. 25
  6. 17
  7. 9409
  8. 10816
  9. 5776
  10. 148
  11. 56
  12. (x + 6)(x + 8)
  13. 36
  14. 441
  15. (x + 8)(x + 9)
  16. (x + 6)(x + 9)
  17. 90
  18. (x + 2)(x + 7)
  19. 49
  20. 3721
  21. 4356
  22. 10
  23. 81
  24. (x + 3)(x + 6)
  25. 185
  26. (x + 2)(x + 3)
  27. (x + 7)(x + 9)
  28. (x + 5)(x + 8)
  29. 1
  30. 25
  31. 625
  32. 225
  33. 3 and 7
  34. 8649
  35. 7225
  36. 324
  37. 5 and 8
  38. 4
  39. 6 and 8
  40. (x + 2)(x + 8)

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