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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 227 · 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 = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 64 (c) 144 (d) 104
  2. 2.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2916 (d) 2116
  5. 5.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 39 (d) 69
  6. 6.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 4 (d) 74
  7. 7.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 4 and 5 (c) 2 and 10 (d) 1 and 8
  8. 8.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 130 (c) 196 (d) 163
  9. 9.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
  10. 10.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 125 (c) 169 (d) 9
  11. 11.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 91 (d) 61
  12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
  13. 13.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)
  14. 14.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 5041 (c) 4771 (d) 4899
  15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 598 (d) 202
  16. 16.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 57 (c) 64 (d) 78
  17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  18. 18.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10816 (d) 10826
  19. 19.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 16 (c) 324 (d) 170
  20. 20.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 36 (d) 12
  21. 21.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 2 and 24 (d) 6 and 8
  22. 22.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 217 (d) 1
  23. 23.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  24. 24.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
  25. 25.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 41 (d) 39
  26. 26.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
  27. 27.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 592 (d) 400
  28. 28.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
  29. 29.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 81 (d) 53
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
  31. 31.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
  32. 32.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 97 (d) 65
  33. 33.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
  34. 34.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 36 (c) 6 (d) 196
  35. 35.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  36. 36.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)
  37. 37.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 9 (d) 10
  38. 38.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 40 (c) 1 and 12 (d) 5 and 8
  39. 39.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 36 (d) 146
  40. 40.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 289 (d) 529

Answer key

  1. 64
  2. (x + 2)(x + 4)
  3. (x + 4)(x + 5)
  4. 2116
  5. 29
  6. 4
  7. 4 and 5
  8. 130
  9. 1849
  10. 81
  11. 1
  12. (x + 2)(x + 9)
  13. (x + 8)(x + 9)
  14. 4761
  15. 202
  16. 50
  17. 72
  18. 10816
  19. 16
  20. 13
  21. 6 and 8
  22. 1
  23. (x + 2)(x + 7)
  24. 11
  25. 40
  26. (x + 3)(x + 5)
  27. 208
  28. 961
  29. 53
  30. 17
  31. 16
  32. 65
  33. 324
  34. 36
  35. (x + 7)(x + 8)
  36. (x + 4)(x + 9)
  37. 9
  38. 5 and 8
  39. 36
  40. 529

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