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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 528 · 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.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
  3. 3.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 184 (d) 104
  4. 4.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
  5. 5.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 100 (d) 16
  6. 6.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 217 (d) 289
  7. 7.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 12 (d) 36
  8. 8.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
  9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  10. 10.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  11. 11.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  13. 13.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
  14. 14.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
  15. 15.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 49 (d) 39
  16. 16.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
  17. 17.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 784 (d) 896
  18. 18.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 97 (d) 133
  19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  20. 20.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4235 (c) 4875 (d) 4225
  21. 21.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  22. 22.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  23. 23.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  24. 24.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10009 (c) 9409 (d) 10609
  25. 25.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 10201 (c) 9811 (d) 9801
  26. 26.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  27. 27.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
  28. 28.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 177 (d) 121
  29. 29.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 85 (c) 157 (d) 121
  30. 30.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2496 (c) 2704 (d) 2304
  31. 31.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3609 (c) 3969 (d) 3979
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  34. 34.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 6 (d) 5
  35. 35.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 18 (d) 1 and 8
  36. 36.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 3 (c) 9 (d) 169
  37. 37.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 1625 (d) 2025
  38. 38.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 2 (d) 164
  39. 39.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 17 (c) 15 (d) 16
  40. 40.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6084 (d) 6404

Answer key

  1. (x + 8)(x + 9)
  2. 6084
  3. 104
  4. (x + 3)(x + 9)
  5. 16
  6. 1
  7. 35
  8. 3249
  9. 16
  10. (x + 5)(x + 6)
  11. 72
  12. (x + 7)(x + 8)
  13. 6 and 7
  14. 15
  15. 29
  16. 5 and 7
  17. 784
  18. 97
  19. 17
  20. 4225
  21. (x + 4)(x + 8)
  22. (x + 2)(x + 3)
  23. (x + 2)(x + 7)
  24. 10609
  25. 9801
  26. (x + 3)(x + 8)
  27. 12
  28. 65
  29. 85
  30. 2304
  31. 3969
  32. (x + 5)(x + 9)
  33. (x + 4)(x + 6)
  34. 6
  35. 3 and 6
  36. 9
  37. 2025
  38. 4
  39. 16
  40. 6724

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