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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 631 · 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 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
  2. 2.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 1 (d) 25
  3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  4. 4.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 1 and 11 (d) 5 and 7
  5. 5.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 9 (c) 2 and 8 (d) 1 and 16
  6. 6.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 4 and 8 (d) 1 and 32
  7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
  8. 8.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  9. 9.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 361 (c) 277 (d) 193
  10. 10.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2401 (d) 2611
  11. 11.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 253 (d) 169
  12. 12.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
  13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  14. 14.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 111 (c) 141 (d) 101
  15. 15.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  16. 16.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 61 (d) 81
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  18. 18.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2491 (d) 2209
  19. 19.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  20. 20.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
  21. 21.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 10 (d) 1 and 30
  22. 22.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 196 (c) 6 (d) 116
  23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
  24. 24.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 225 (c) 5 (d) 125
  25. 25.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 4 (c) 400 (d) 208
  26. 26.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1764 (d) 1444
  27. 27.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
  28. 28.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4489 (d) 4891
  29. 29.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 13 (c) 16 (d) 22
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 6416 (d) 7066
  32. 32.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
  34. 34.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 169 (c) 3 (d) 89
  35. 35.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 10
  36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  37. 37.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 15 (c) 10 (d) 16
  38. 38.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904
  39. 39.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
  40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 229 (d) 409

Answer key

  1. 9801
  2. 1
  3. (x + 7)(x + 9)
  4. 5 and 7
  5. 2 and 8
  6. 4 and 8
  7. (x + 3)(x + 5)
  8. (x + 5)(x + 9)
  9. 193
  10. 2601
  11. 85
  12. (x + 4)(x + 8)
  13. (x + 5)(x + 7)
  14. 101
  15. (x + 5)(x + 8)
  16. 41
  17. 17
  18. 2209
  19. (x + 6)(x + 7)
  20. 72
  21. 5 and 6
  22. 36
  23. 13
  24. 25
  25. 16
  26. 1444
  27. 484
  28. 4489
  29. 10
  30. (x + 7)(x + 8)
  31. 7056
  32. (x + 4)(x + 7)
  33. 1089
  34. 9
  35. 2 and 9
  36. 15
  37. 16
  38. 1024
  39. 42
  40. 169

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