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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 54 · 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 = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
  2. 2.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 172 (d) 244
  3. 3.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
  4. 4.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 36 (d) 100
  5. 5.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 136 (d) 4
  6. 6.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 1 and 40 (d) 5 and 8
  7. 7.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 25 (d) 361
  8. 8.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
  9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
  10. 10.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 81 (c) 41 (d) 61
  11. 11.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  12. 12.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
  13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
  14. 14.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 441 (d) 401
  16. 16.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 2 and 18 (d) 1 and 12
  17. 17.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 14 (d) 16
  18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 794 (d) 896
  21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
  22. 22.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
  23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  24. 24.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1521 (c) 1691 (d) 1601
  25. 25.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 9 (d) 20
  26. 26.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 22 (d) 10
  27. 27.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  28. 28.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 56 (d) 14
  29. 29.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  30. 30.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  31. 31.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 3035 (d) 2525
  32. 32.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7569 (d) 7579
  33. 33.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
  34. 34.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 117 (d) 3
  35. 35.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 82 (c) 91 (d) 118
  36. 36.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 235 (d) 625
  37. 37.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 9 (d) 10
  38. 38.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 13 (d) 15
  39. 39.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 289 (d) 391
  40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)

Answer key

  1. 180
  2. 148
  3. 13
  4. 36
  5. 16
  6. 5 and 8
  7. 25
  8. 8464
  9. (x + 3)(x + 8)
  10. 1
  11. 42
  12. 35
  13. 17
  14. (x + 4)(x + 8)
  15. 441
  16. 4 and 9
  17. 15
  18. (x + 2)(x + 7)
  19. (x + 5)(x + 6)
  20. 784
  21. 72
  22. 4
  23. (x + 6)(x + 7)
  24. 1681
  25. 20
  26. 21
  27. (x + 8)(x + 9)
  28. 15
  29. (x + 5)(x + 7)
  30. 48
  31. 3025
  32. 7569
  33. 16
  34. 9
  35. 82
  36. 225
  37. 10
  38. 14
  39. 289
  40. (x + 7)(x + 9)

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