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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 668 · 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.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  2. 2.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904
  3. 3.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 875 (c) 1225 (d) 635
  4. 4.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  7. 7.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 11 (d) 10
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  9. 9.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 19 (d) 21
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  11. 11.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
  13. 13.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 5 (c) 7 (d) 1
  14. 14.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 10 (d) 11
  15. 15.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 223 (d) 421
  16. 16.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
  17. 17.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3481 (d) 3491
  18. 18.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 10 (d) 28
  19. 19.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 7744 (d) 8464
  20. 20.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 148 (c) 76 (d) 100
  21. 21.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 14 (c) 9 (d) 15
  22. 22.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2401 (d) 2601
  23. 23.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 21 (d) 3 and 7
  24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  25. 25.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  26. 26.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
  27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  28. 28.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)
  29. 29.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 2 and 10 (d) 1 and 11
  30. 30.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8116 (c) 8836 (d) 8846
  31. 31.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
  32. 32.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 8 (c) 7 (d) 5
  33. 33.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6094 (d) 6396
  34. 34.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 65 (c) 121 (d) 93
  35. 35.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3854 (c) 3364 (d) 3604
  36. 36.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 64 (c) 4 (d) 16
  37. 37.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 196 (d) 100
  38. 38.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2126 (d) 2916
  40. 40.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37

Answer key

  1. 8
  2. 1024
  3. 625
  4. (x + 2)(x + 4)
  5. (x + 8)(x + 9)
  6. (x + 7)(x + 8)
  7. 11
  8. 17
  9. 20
  10. 16
  11. (x + 3)(x + 9)
  12. 5625
  13. 1
  14. 11
  15. 157
  16. 25
  17. 3481
  18. 11
  19. 8464
  20. 52
  21. 14
  22. 2401
  23. 3 and 7
  24. (x + 6)(x + 7)
  25. (x + 4)(x + 6)
  26. 1681
  27. (x + 4)(x + 8)
  28. (x + 5)(x + 6)
  29. 5 and 7
  30. 8836
  31. 10816
  32. 6
  33. 6084
  34. 65
  35. 3844
  36. 16
  37. 100
  38. (x + 2)(x + 8)
  39. 2116
  40. 1

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