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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 398 · 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 54²? (a) 2516 (b) 2926 (c) 2916 (d) 2116
  2. 2.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 904 (d) 1024
  3. 3.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 361 (d) 399
  4. 4.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)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 289 (d) 529
  6. 6.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
  7. 7.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6094 (c) 6724 (d) 6396
  8. 8.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 15 (d) 57
  9. 9.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
  10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  11. 11.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 229 (d) 409
  12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  13. 13.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 2 and 7 (c) 1 and 14 (d) 3 and 6
  14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  15. 15.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 5476 (d) 4366
  16. 16.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
  17. 17.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 10 (c) 20 (d) 21
  18. 18.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
  19. 19.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 153 (d) 225
  20. 20.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
  21. 21.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10816 (c) 10016 (d) 9216
  22. 22.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
  23. 23.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 117 (d) 45
  24. 24.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 145 (d) 433
  25. 25.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 82 (c) 100 (d) 91
  26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  27. 27.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 89 (d) 129
  28. 28.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  29. 29.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 14 (d) 16
  30. 30.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 3 and 15 (d) 1 and 45
  31. 31.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
  33. 33.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 19 (c) 1 (d) 13
  34. 34.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3969 (c) 3249 (d) 3591
  35. 35.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1369 (d) 1849
  36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  37. 37.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 1 and 10 (d) 5 and 6
  38. 38.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 74 (c) 214 (d) 109
  39. 39.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 61 (d) 91
  40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)

Answer key

  1. 2916
  2. 1024
  3. 361
  4. (x + 8)(x + 9)
  5. 289
  6. 225
  7. 6084
  8. 56
  9. 12
  10. (x + 7)(x + 8)
  11. 169
  12. 72
  13. 2 and 7
  14. 1
  15. 4356
  16. 208
  17. 21
  18. (x + 3)(x + 4)
  19. 81
  20. 180
  21. 10816
  22. 7 and 8
  23. 45
  24. 145
  25. 82
  26. (x + 6)(x + 7)
  27. 89
  28. 10
  29. 15
  30. 5 and 9
  31. (x + 7)(x + 9)
  32. 625
  33. 1
  34. 3249
  35. 1849
  36. (x + 6)(x + 8)
  37. 5 and 6
  38. 74
  39. 61
  40. (x + 5)(x + 8)

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