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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 220 · 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 = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  2. 2.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 106 (c) 4 (d) 196
  3. 3.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7235 (c) 6425 (d) 7225
  5. 5.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2126 (d) 2484
  6. 6.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 9 (d) 7
  7. 7.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 841 (d) 899
  8. 8.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1764 (d) 1444
  11. 11.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 80 (c) 4 (d) 144
  12. 12.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
  13. 13.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 13 (d) 12
  14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  15. 15.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
  16. 16.What is the sum of the roots of x² − 13x + 36 = 0? (a) 13 (b) 14 (c) 12 (d) 36
  17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  18. 18.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 46 (d) 14
  19. 19.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 49 (d) 37
  20. 20.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 2 and 12 (d) 1 and 24
  21. 21.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
  22. 22.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15
  23. 23.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 32 (d) 33
  24. 24.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6241 (d) 6399
  25. 25.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 1 and 10 (c) 2 and 5 (d) 3 and 4
  26. 26.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 4624 (d) 5184
  27. 27.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  28. 28.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 8 (d) 10
  29. 29.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
  30. 30.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  31. 31.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 117
  32. 32.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 1 and 15 (c) 2 and 6 (d) 3 and 5
  33. 33.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 67 (d) 53
  34. 34.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  35. 35.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
  36. 36.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  37. 37.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
  38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 4225 (d) 3025
  39. 39.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 223 (d) 157
  40. 40.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 4 (c) 208 (d) 400

Answer key

  1. 1
  2. 16
  3. (x + 3)(x + 5)
  4. 7225
  5. 2116
  6. 8
  7. 841
  8. 63
  9. (x + 3)(x + 6)
  10. 1444
  11. 16
  12. 7 and 8
  13. 12
  14. 17
  15. 4
  16. 13
  17. (x + 6)(x + 7)
  18. 45
  19. 25
  20. 4 and 6
  21. 5929
  22. 56
  23. 32
  24. 6241
  25. 2 and 5
  26. 4624
  27. 16
  28. 9
  29. (x + 2)(x + 6)
  30. 6
  31. 9
  32. 3 and 5
  33. 53
  34. 181
  35. 1849
  36. (x + 5)(x + 6)
  37. 125
  38. 3025
  39. 157
  40. 16

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