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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 376 · 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 = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 36 (c) 6 (d) 256
  2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 145 (d) 217
  3. 3.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 124 (d) 104
  4. 4.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 971 (c) 901 (d) 841
  5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  6. 6.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 4 and 5 (d) 2 and 10
  7. 7.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 2 and 5 (c) 3 and 4 (d) 1 and 10
  8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  9. 9.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 133 (c) 144 (d) 122
  10. 10.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  11. 11.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 81 (d) 9
  12. 12.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 4 (c) 40 (d) 64
  13. 13.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  14. 14.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  15. 15.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10414 (d) 10404
  16. 16.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 13 (d) 12
  17. 17.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  18. 18.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
  19. 19.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
  20. 20.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 24 (d) 11
  21. 21.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 15 (d) 13
  22. 22.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 6 (c) 8 (d) 12
  23. 23.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 208 (d) 160
  24. 24.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
  25. 25.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8099 (c) 8281 (d) 7921
  26. 26.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2496 (d) 2314
  27. 27.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 30 (d) 10
  28. 28.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 31 (d) 26
  29. 29.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  31. 31.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 313 (c) 137 (d) 225
  32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
  33. 33.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
  34. 34.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  36. 36.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  37. 37.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 11 (c) 7 (d) 13
  38. 38.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 117 (d) 333
  39. 39.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 169 (d) 253
  40. 40.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 64 (c) 36 (d) 50

Answer key

  1. 36
  2. 1
  3. 104
  4. 961
  5. (x + 7)(x + 8)
  6. 4 and 5
  7. 2 and 5
  8. (x + 5)(x + 6)
  9. 122
  10. 56
  11. 9
  12. 16
  13. (x + 5)(x + 7)
  14. 61
  15. 10404
  16. 12
  17. (x + 3)(x + 6)
  18. 8 and 9
  19. 361
  20. 11
  21. 14
  22. 7
  23. 160
  24. 13
  25. 7921
  26. 2304
  27. 11
  28. 26
  29. (x + 6)(x + 9)
  30. 8836
  31. 137
  32. 202
  33. 576
  34. 7225
  35. (x + 8)(x + 9)
  36. (x + 4)(x + 5)
  37. 12
  38. 117
  39. 85
  40. 36

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