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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 494 · 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 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
  2. 2.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
  3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
  5. 5.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  6. 6.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  8. 8.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 5 (d) 7
  9. 9.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
  10. 10.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 4356 (d) 5476
  11. 11.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  12. 12.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  13. 13.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 26 (d) 31
  14. 14.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
  15. 15.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 14 (d) 15
  16. 16.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 10 (d) 11
  18. 18.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  19. 19.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 6 (d) 146
  20. 20.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 12
  21. 21.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  22. 22.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 10 (c) 144 (d) 100
  23. 23.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
  24. 24.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 34 (c) 35 (d) 12
  25. 25.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 8 (d) 6
  26. 26.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 9 (d) 7
  27. 27.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 1 and 28 (c) 2 and 14 (d) 4 and 7
  28. 28.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1454 (c) 1596 (d) 1444
  29. 29.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 1 and 14 (c) 1 and 8 (d) 2 and 7
  30. 30.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  31. 31.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)
  32. 32.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 28 (d) 12
  33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  34. 34.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 4 and 8 (c) 2 and 16 (d) 1 and 11
  35. 35.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 196
  36. 36.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
  37. 37.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 1 and 10 (d) 2 and 15
  38. 38.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
  39. 39.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 884 (c) 676 (d) 686
  40. 40.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4916 (c) 4356 (d) 5486

Answer key

  1. 2304
  2. 256
  3. (x + 2)(x + 3)
  4. (x + 5)(x + 7)
  5. (x + 6)(x + 9)
  6. (x + 6)(x + 8)
  7. (x + 8)(x + 9)
  8. 5
  9. 63
  10. 4356
  11. 8 and 9
  12. 17
  13. 26
  14. (x + 2)(x + 9)
  15. 14
  16. (x + 5)(x + 9)
  17. 11
  18. (x + 2)(x + 6)
  19. 36
  20. 11
  21. 16
  22. 100
  23. 73
  24. 35
  25. 6
  26. 1
  27. 4 and 7
  28. 1444
  29. 2 and 7
  30. 10404
  31. (x + 4)(x + 5)
  32. 11
  33. (x + 7)(x + 8)
  34. 4 and 8
  35. 4
  36. 324
  37. 5 and 6
  38. 12
  39. 676
  40. 5476

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