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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 233 · 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 = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 116 (d) 156
  2. 2.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 67 (c) 81 (d) 109
  3. 3.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
  4. 4.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 81 (d) 153
  5. 5.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
  6. 6.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  7. 7.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 9 (d) 6
  8. 8.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
  9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 17 (d) 15
  10. 10.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 598 (d) 400
  11. 11.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 18 (d) 9
  12. 12.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
  13. 13.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 61 (d) 121
  14. 14.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 41 (d) 13
  15. 15.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4761 (c) 5051 (d) 4901
  16. 16.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 8084 (c) 7406 (d) 7396
  17. 17.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 64 (d) 63
  18. 18.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
  19. 19.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  20. 20.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 3 (d) 9
  21. 21.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 10
  22. 22.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)
  23. 23.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 4 (d) 100
  24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
  26. 26.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 82 (c) 100 (d) 118
  27. 27.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4925 (d) 4225
  28. 28.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 49 (c) 7 (d) 85
  29. 29.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8474 (c) 8104 (d) 7744
  31. 31.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
  32. 32.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 121 (c) 103 (d) 85
  33. 33.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  34. 34.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  35. 35.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 9 (d) 24
  36. 36.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3491 (d) 3721
  37. 37.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 256 (d) 4
  38. 38.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
  39. 39.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
  40. 40.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 9 (c) 14 (d) 8

Answer key

  1. 116
  2. 53
  3. 2704
  4. 81
  5. 1849
  6. (x + 4)(x + 9)
  7. 8
  8. 3 and 5
  9. 16
  10. 202
  11. 9
  12. 225
  13. 61
  14. 42
  15. 5041
  16. 7396
  17. 63
  18. 7 and 9
  19. 6561
  20. 9
  21. 24
  22. (x + 6)(x + 7)
  23. 4
  24. 181
  25. (x + 2)(x + 5)
  26. 82
  27. 5625
  28. 49
  29. (x + 2)(x + 7)
  30. 8464
  31. 9216
  32. 85
  33. (x + 2)(x + 8)
  34. (x + 4)(x + 8)
  35. 10
  36. 3481
  37. 4
  38. 12
  39. 1
  40. 9

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