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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 948 · 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.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10404 (d) 10414
  2. 2.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3604 (c) 3844 (d) 3364
  3. 3.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 202 (c) 301 (d) 598
  4. 4.If a + b = 15 and ab = 36, what is a² + b²? (a) 189 (b) 297 (c) 225 (d) 153
  5. 5.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 169 (c) 145 (d) 193
  6. 6.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 1 (d) 91
  7. 7.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
  8. 8.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 5 (c) 97 (d) 169
  9. 9.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 47 (d) 14
  10. 10.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
  11. 11.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 5939 (d) 6391
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  13. 13.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
  14. 14.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 784 (d) 896
  15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  16. 16.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 3 and 5 (d) 1 and 7
  17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  18. 18.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  20. 20.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
  21. 21.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 8 (c) 9 (d) 18
  22. 22.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 79 (d) 100
  23. 23.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
  24. 24.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 7 and 8 (d) 1 and 14
  25. 25.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 4
  26. 26.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1369 (c) 1859 (d) 1609
  27. 27.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 7744 (d) 8464
  28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  29. 29.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 4 and 5 (d) 1 and 20
  30. 30.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 384 (d) 256
  31. 31.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  32. 32.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
  33. 33.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 225 (d) 5
  34. 34.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 40 (d) 13
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
  36. 36.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 971 (d) 901
  37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  38. 38.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 441
  39. 39.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
  40. 40.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 3 (d) 149

Answer key

  1. 10404
  2. 3844
  3. 202
  4. 153
  5. 145
  6. 1
  7. 1444
  8. 25
  9. 48
  10. 11
  11. 5929
  12. (x + 6)(x + 7)
  13. 17
  14. 784
  15. (x + 4)(x + 8)
  16. 3 and 5
  17. (x + 4)(x + 7)
  18. (x + 7)(x + 9)
  19. (x + 5)(x + 9)
  20. 164
  21. 9
  22. 58
  23. (x + 5)(x + 8)
  24. 7 and 8
  25. 5
  26. 1849
  27. 7744
  28. (x + 8)(x + 9)
  29. 4 and 5
  30. 256
  31. (x + 5)(x + 6)
  32. 6
  33. 25
  34. 40
  35. 16
  36. 961
  37. 181
  38. 9
  39. 11
  40. 9

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