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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 474 · 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 35²? (a) 1235 (b) 1225 (c) 2025 (d) 1575
  2. 2.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 657 (d) 441
  3. 3.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  4. 4.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
  5. 5.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 256 (d) 193
  6. 6.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 16 (c) 100 (d) 4
  7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  8. 8.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 441 (d) 225
  9. 9.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  10. 10.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 4 (c) 80 (d) 144
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  12. 12.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 256 (d) 576
  14. 14.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 25 (c) 73 (d) 37
  15. 15.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 409 (d) 529
  16. 16.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 289 (c) 157 (d) 25
  17. 17.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 399 (d) 361
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1591 (d) 1369
  21. 21.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
  22. 22.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2209 (c) 2219 (d) 2809
  23. 23.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3604 (d) 3854
  24. 24.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 9999 (d) 10201
  25. 25.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 9 (d) 225
  26. 26.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4225 (c) 5625 (d) 4925
  27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 9975 (d) 11025
  29. 29.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 36 (c) 13 (d) 37
  30. 30.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 10609 (d) 9409
  31. 31.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 6 (d) 9
  32. 32.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 10 (d) 16
  33. 33.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 10 (d) 23
  34. 34.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 5 and 7 (c) 2 and 10 (d) 1 and 35
  35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  36. 36.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 6 and 8 (c) 1 and 13 (d) 2 and 24
  37. 37.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  39. 39.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 85 (d) 127
  40. 40.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14

Answer key

  1. 1225
  2. 225
  3. (x + 2)(x + 5)
  4. 13
  5. 130
  6. 16
  7. (x + 5)(x + 8)
  8. 9
  9. (x + 5)(x + 7)
  10. 16
  11. (x + 8)(x + 9)
  12. (x + 4)(x + 6)
  13. 576
  14. 25
  15. 529
  16. 25
  17. 361
  18. (x + 2)(x + 8)
  19. (x + 2)(x + 6)
  20. 1369
  21. 100
  22. 2209
  23. 3844
  24. 9801
  25. 9
  26. 5625
  27. (x + 3)(x + 9)
  28. 9025
  29. 36
  30. 10609
  31. 8
  32. 10
  33. 24
  34. 5 and 7
  35. (x + 3)(x + 4)
  36. 6 and 8
  37. 729
  38. 17
  39. 1
  40. 15

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