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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 851 · 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 = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  2. 2.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 28 (c) 29 (d) 11
  3. 3.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  4. 4.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 11 (c) 7 (d) 13
  5. 5.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10001 (c) 9801 (d) 10201
  6. 6.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 170 (d) 4
  7. 7.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 11025 (c) 9975 (d) 9025
  8. 8.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1859 (d) 1849
  9. 9.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7225 (d) 7235
  10. 10.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 80 (c) 208 (d) 144
  11. 11.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
  12. 12.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1764 (d) 1454
  13. 13.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 28 (d) 11
  14. 14.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904
  15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  16. 16.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 64 (d) 34
  17. 17.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 39 (c) 29 (d) 69
  18. 18.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
  19. 19.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
  20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  21. 21.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
  22. 22.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 841 (d) 961
  23. 23.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  24. 24.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 12 (c) 32 (d) 13
  25. 25.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  27. 27.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
  28. 28.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
  30. 30.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 8 (d) 7
  31. 31.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 36 (c) 50 (d) 6
  32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
  33. 33.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 3616 (c) 4106 (d) 3136
  34. 34.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3584 (c) 3136 (d) 3146
  35. 35.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 157 (c) 169 (d) 145
  36. 36.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 676 (d) 1166
  37. 37.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
  38. 38.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
  39. 39.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
  40. 40.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 57 (c) 50 (d) 78

Answer key

  1. 1
  2. 28
  3. 10
  4. 12
  5. 10201
  6. 16
  7. 9025
  8. 1849
  9. 7225
  10. 80
  11. 180
  12. 1444
  13. 11
  14. 1024
  15. (x + 6)(x + 9)
  16. 34
  17. 29
  18. 63
  19. 10
  20. (x + 2)(x + 4)
  21. 56
  22. 841
  23. (x + 6)(x + 7)
  24. 12
  25. 5476
  26. (x + 8)(x + 9)
  27. (x + 5)(x + 6)
  28. (x + 5)(x + 9)
  29. 961
  30. 8
  31. 36
  32. 1369
  33. 4096
  34. 3136
  35. 145
  36. 1156
  37. 1
  38. 2304
  39. 7056
  40. 50

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