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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 465 · 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 − 10x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 10
  2. 2.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 12 (c) 2 and 21 (d) 1 and 42
  3. 3.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2116 (d) 2126
  4. 4.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 49 (d) 7
  5. 5.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
  6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  8. 8.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  9. 9.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2916 (d) 2926
  10. 10.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 8099 (d) 7931
  11. 11.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 409 (d) 289
  12. 12.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 112 (c) 208 (d) 144
  13. 13.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  14. 14.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  15. 15.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 33 (d) 31
  16. 16.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 10 (d) 2 and 5
  17. 17.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 40 (d) 52
  18. 18.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
  19. 19.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2704 (c) 2304 (d) 2714
  21. 21.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 125 (d) 225
  22. 22.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7
  23. 23.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 16 (d) 136
  24. 24.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9996 (d) 9604
  25. 25.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 36 (c) 324 (d) 6
  26. 26.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
  27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 916 (d) 1166
  28. 28.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
  29. 29.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 4 (d) 2
  30. 30.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 193 (d) 382
  31. 31.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
  32. 32.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
  33. 33.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 101 (c) 111 (d) 141
  34. 34.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2
  35. 35.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3136 (c) 4096 (d) 3616
  36. 36.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  37. 37.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  38. 38.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 10816 (d) 9984
  39. 39.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)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 925 (d) 625

Answer key

  1. 24
  2. 6 and 7
  3. 2116
  4. 49
  5. 13
  6. (x + 6)(x + 9)
  7. 2025
  8. 181
  9. 2916
  10. 7921
  11. 529
  12. 80
  13. 1
  14. (x + 4)(x + 5)
  15. 32
  16. 2 and 5
  17. 40
  18. 193
  19. (x + 5)(x + 7)
  20. 2704
  21. 25
  22. 8
  23. 16
  24. 9604
  25. 36
  26. 6889
  27. 1156
  28. 49
  29. 4
  30. 130
  31. 45
  32. 225
  33. 101
  34. 4
  35. 4096
  36. (x + 2)(x + 6)
  37. (x + 4)(x + 8)
  38. 9216
  39. (x + 5)(x + 8)
  40. 1225

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