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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 319 · 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.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 5 (c) 25 (d) 121
  2. 2.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 8
  3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
  4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 625 (d) 235
  6. 6.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 28 (d) 1 and 10
  7. 7.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 16 (d) 2 and 8
  8. 8.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 73 (c) 49 (d) 37
  9. 9.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
  10. 10.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 97 (c) 121 (d) 73
  11. 11.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 12 (d) 8
  12. 12.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 147 (c) 213 (d) 125
  13. 13.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
  14. 14.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2916 (d) 2484
  15. 15.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 100 (c) 148 (d) 196
  16. 16.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
  17. 17.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 91 (d) 82
  18. 18.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2401 (c) 2411 (d) 2499
  19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2516 (c) 2116 (d) 2916
  21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2401 (d) 2601
  23. 23.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
  24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  25. 25.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 145 (d) 289
  26. 26.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3979 (d) 3969
  27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  28. 28.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 400 (d) 208
  29. 29.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
  30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6241 (d) 6571
  31. 31.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3969 (c) 3259 (d) 3249
  32. 32.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 10 (c) 9 (d) 20
  33. 33.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)
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  35. 35.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
  36. 36.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 144 (c) 133 (d) 166
  37. 37.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 1 and 10 (c) 3 and 4 (d) 2 and 5
  38. 38.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1225 (d) 1235
  39. 39.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 851 (d) 961

Answer key

  1. 25
  2. 6
  3. 1936
  4. (x + 4)(x + 6)
  5. 225
  6. 4 and 7
  7. 2 and 8
  8. 25
  9. 16
  10. 73
  11. 8
  12. 125
  13. 5 and 6
  14. 2116
  15. 100
  16. 8 and 9
  17. 82
  18. 2401
  19. (x + 6)(x + 9)
  20. 2916
  21. (x + 7)(x + 9)
  22. 2601
  23. (x + 4)(x + 7)
  24. (x + 2)(x + 5)
  25. 1
  26. 3969
  27. (x + 3)(x + 9)
  28. 16
  29. 20
  30. 6561
  31. 3249
  32. 9
  33. (x + 8)(x + 9)
  34. 17
  35. 7 and 9
  36. 122
  37. 2 and 5
  38. 1225
  39. (x + 4)(x + 8)
  40. 841

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