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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 12 · 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 = 30, what is a² + b²? (a) 181 (b) 121 (c) 61 (d) 91
  2. 2.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1596 (d) 1454
  3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
  4. 4.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 52
  5. 5.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  6. 6.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 13 (d) 12
  7. 7.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
  8. 8.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
  9. 9.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 48 (d) 1 and 13
  10. 10.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 171 (c) 225 (d) 333
  11. 11.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 2 and 12 (d) 1 and 24
  12. 12.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 253 (c) 127 (d) 169
  13. 13.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
  14. 14.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 19 (d) 9
  15. 15.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
  16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4884 (d) 4356
  18. 18.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
  19. 19.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 53 (c) 25 (d) 5
  20. 20.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 289 (d) 217
  21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
  22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 9999 (d) 10201
  23. 23.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 1 (d) 121
  24. 24.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
  25. 25.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
  26. 26.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4904 (c) 5184 (d) 4624
  27. 27.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 5 and 6 (d) 1 and 30
  28. 28.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 153 (c) 297 (d) 189
  29. 29.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 3 (c) 117 (d) 225
  30. 30.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 9 (d) 10
  31. 31.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 11 (d) 23
  32. 32.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 141 (c) 111 (d) 101
  33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
  34. 34.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  35. 35.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
  36. 36.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1936 (d) 1946
  37. 37.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 25 (c) 5 (d) 97
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 441
  39. 39.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 193 (d) 25
  40. 40.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 36 (d) 4

Answer key

  1. 61
  2. 1444
  3. 181
  4. 4
  5. (x + 5)(x + 9)
  6. 12
  7. 1
  8. 8 and 9
  9. 6 and 8
  10. 117
  11. 4 and 6
  12. 85
  13. 1
  14. 20
  15. (x + 2)(x + 5)
  16. (x + 6)(x + 9)
  17. 4356
  18. 7
  19. 25
  20. 1
  21. 145
  22. 9801
  23. 1
  24. 3721
  25. 4
  26. 5184
  27. 5 and 6
  28. 153
  29. 9
  30. 10
  31. 24
  32. 101
  33. 225
  34. (x + 7)(x + 9)
  35. 10
  36. 1936
  37. 25
  38. 9
  39. 25
  40. 4

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