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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 190 · 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 sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
  2. 2.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 16 (d) 15
  3. 3.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
  4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
  5. 5.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 2 (d) 196
  6. 6.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1379 (c) 1369 (d) 1849
  7. 7.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5486 (c) 4916 (d) 5476
  8. 8.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 68 (c) 6 (d) 36
  9. 9.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
  10. 10.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 292 (c) 100 (d) 196
  11. 11.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 13 (d) 12
  12. 12.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
  13. 13.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
  14. 14.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 1 and 9 (c) 3 and 7 (d) 2 and 8
  15. 15.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 225 (b) 375 (c) 235 (d) 625
  16. 16.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9226 (d) 9984
  17. 17.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 43 (d) 13
  18. 18.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
  19. 19.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 25 (d) 24
  20. 20.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10816 (c) 9216 (d) 10826
  21. 21.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2916 (c) 2116 (d) 2926
  22. 22.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 5 and 8 (d) 1 and 40
  23. 23.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 25 (c) 73 (d) 37
  24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  25. 25.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  26. 26.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
  27. 27.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 133 (d) 122
  28. 28.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 11 (d) 13
  29. 29.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
  30. 30.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 90 (d) 198
  31. 31.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 400 (d) 301
  33. 33.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
  34. 34.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 20 (c) 36 (d) 2
  35. 35.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 4 (d) 144
  36. 36.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 371 (d) 399
  37. 37.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6899 (d) 6889
  38. 38.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1859 (c) 1369 (d) 1609
  39. 39.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)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3969 (c) 3249 (d) 3609

Answer key

  1. 13
  2. 15
  3. 193
  4. 56
  5. 4
  6. 1369
  7. 5476
  8. 36
  9. 164
  10. 100
  11. 12
  12. 16
  13. 30
  14. 3 and 7
  15. 225
  16. 9216
  17. 42
  18. (x + 4)(x + 6)
  19. 24
  20. 10816
  21. 2916
  22. 5 and 8
  23. 25
  24. (x + 6)(x + 7)
  25. (x + 5)(x + 6)
  26. (x + 3)(x + 5)
  27. 122
  28. 12
  29. 961
  30. 90
  31. (x + 7)(x + 8)
  32. 202
  33. 1
  34. 4
  35. 4
  36. 361
  37. 6889
  38. 1849
  39. (x + 8)(x + 9)
  40. 3969

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