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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 640 · 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.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 13 (d) 41
  2. 2.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
  3. 3.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1306 (c) 1296 (d) 1584
  4. 4.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  5. 5.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 17 (d) 9
  6. 6.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)
  7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 196 (d) 276
  8. 8.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  9. 9.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
  10. 10.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 256 (c) 266 (d) 384
  11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  12. 12.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 851 (c) 961 (d) 899
  14. 14.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
  15. 15.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
  16. 16.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2401 (c) 2601 (d) 2499
  17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  18. 18.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 529 (d) 193
  19. 19.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3969 (c) 3979 (d) 3249
  20. 20.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 6 (c) 2 and 4 (d) 1 and 4
  21. 21.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2491 (d) 2809
  22. 22.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
  23. 23.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400
  24. 24.If a + b = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 65 (c) 121 (d) 3
  25. 25.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6409 (d) 6899
  26. 26.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 20 (d) 19
  27. 27.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2314 (c) 2704 (d) 2304
  29. 29.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 145 (d) 289
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 10201 (d) 9801
  32. 32.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 5041 (d) 4761
  33. 33.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 4 (c) 16 (d) 58
  34. 34.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 9 (c) 18 (d) 8
  35. 35.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 129 (d) 249
  36. 36.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 1
  37. 37.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1591 (d) 1849
  38. 38.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 67 (c) 109 (d) 53
  39. 39.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6241 (d) 6561
  40. 40.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21

Answer key

  1. 42
  2. 16
  3. 1296
  4. (x + 4)(x + 9)
  5. 9
  6. (x + 8)(x + 9)
  7. 116
  8. (x + 2)(x + 4)
  9. 4
  10. 256
  11. (x + 7)(x + 9)
  12. (x + 3)(x + 9)
  13. 841
  14. 202
  15. 16
  16. 2401
  17. (x + 5)(x + 8)
  18. 193
  19. 3969
  20. 2 and 3
  21. 2209
  22. 148
  23. 208
  24. 9
  25. 6889
  26. 20
  27. (x + 4)(x + 8)
  28. 2304
  29. 1
  30. (x + 7)(x + 8)
  31. 9801
  32. 4761
  33. 16
  34. 9
  35. 89
  36. 1
  37. 1369
  38. 53
  39. 6561
  40. 7 and 9

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