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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 176 · 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 54²? (a) 2516 (b) 2916 (c) 2116 (d) 2926
  2. 2.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 43 (d) 37
  3. 3.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324
  4. 4.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 2 and 28 (d) 1 and 56
  5. 5.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 784 (d) 1034
  6. 6.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 2 and 6 (d) 1 and 15
  7. 7.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)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6561 (d) 6241
  9. 9.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 97 (c) 133 (d) 169
  10. 10.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
  11. 11.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 4 (d) 10
  12. 12.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 97 (c) 81 (d) 65
  13. 13.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 53 (d) 109
  14. 14.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 169 (d) 89
  15. 15.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 2 (d) 4
  16. 16.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 146 (d) 36
  17. 17.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
  18. 18.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5635 (c) 5625 (d) 7225
  19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
  21. 21.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 121 (c) 11 (d) 169
  22. 22.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 9 (d) 11
  23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
  24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
  25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
  26. 26.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 8 (d) 14
  27. 27.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
  28. 28.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
  29. 29.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 794 (d) 896
  30. 30.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)
  31. 31.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 213 (c) 147 (d) 169
  32. 32.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 361 (d) 181
  33. 33.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
  34. 34.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 9 (c) 10 (d) 8
  35. 35.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 37 (d) 25
  36. 36.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 2 and 12 (d) 4 and 6
  37. 37.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  38. 38.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 6
  39. 39.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
  40. 40.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)

Answer key

  1. 2916
  2. 37
  3. 324
  4. 7 and 8
  5. 1024
  6. 3 and 5
  7. (x + 3)(x + 5)
  8. 6241
  9. 97
  10. 54
  11. 4
  12. 65
  13. 53
  14. 89
  15. 4
  16. 36
  17. 841
  18. 5625
  19. 16
  20. 145
  21. 121
  22. 10
  23. 25
  24. (x + 4)(x + 7)
  25. (x + 2)(x + 7)
  26. 15
  27. 7
  28. 9
  29. 784
  30. (x + 6)(x + 7)
  31. 125
  32. 181
  33. (x + 5)(x + 8)
  34. 9
  35. 25
  36. 4 and 6
  37. (x + 2)(x + 5)
  38. 7
  39. 1444
  40. (x + 2)(x + 4)

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