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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 610 · 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) 2916 (c) 2116 (d) 2926
  2. 2.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 9025 (d) 8075
  3. 3.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 13 (d) 12
  4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 244 (d) 484
  5. 5.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 7921 (d) 8099
  6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
  8. 8.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3844 (d) 3596
  9. 9.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 144 (c) 133 (d) 166
  10. 10.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 4 (d) 324
  11. 11.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2496 (c) 2314 (d) 2304
  12. 12.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 11025 (d) 9025
  15. 15.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
  16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  17. 17.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  18. 18.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 28 (d) 36
  19. 19.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 21 (d) 20
  20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3481 (d) 3721
  21. 21.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1379 (c) 1369 (d) 1591
  22. 22.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 41 (d) 42
  23. 23.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 4761 (d) 5051
  24. 24.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8104 (d) 8464
  25. 25.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 193 (c) 361 (d) 529
  27. 27.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 9 (d) 3
  28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4489 (c) 5329 (d) 4891
  30. 30.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 1225 (d) 2025
  31. 31.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
  32. 32.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 2 and 5 (c) 1 and 10 (d) 3 and 4
  33. 33.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
  34. 34.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
  35. 35.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1936 (d) 1946
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 45 (d) 117
  37. 37.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 225 (d) 1
  38. 38.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 100 (c) 79 (d) 142
  39. 39.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 24 (c) 11 (d) 10
  40. 40.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9604 (c) 9614 (d) 9996

Answer key

  1. 2916
  2. 7225
  3. 13
  4. 164
  5. 7921
  6. (x + 5)(x + 6)
  7. 441
  8. 3364
  9. 122
  10. 4
  11. 2304
  12. (x + 2)(x + 5)
  13. (x + 8)(x + 9)
  14. 11025
  15. 7 and 8
  16. (x + 4)(x + 9)
  17. (x + 4)(x + 6)
  18. 20
  19. 21
  20. 3481
  21. 1369
  22. 42
  23. 5041
  24. 8464
  25. 72
  26. 193
  27. 9
  28. (x + 7)(x + 9)
  29. 4489
  30. 1225
  31. 7396
  32. 2 and 5
  33. 484
  34. 7
  35. 1936
  36. 45
  37. 1
  38. 58
  39. 10
  40. 9604

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