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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 919 · 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 − 7x + 10 = 0? (a) 10 (b) 7 (c) 11 (d) 9
  2. 2.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
  3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
  5. 5.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 9 (d) 11
  6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 202 (d) 598
  7. 7.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 11 (d) 13
  8. 8.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 841 (d) 961
  10. 10.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 6391 (d) 5929
  11. 11.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 4634 (d) 5184
  12. 12.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
  13. 13.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 225 (c) 425 (d) 625
  14. 14.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
  15. 15.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5339 (d) 5329
  16. 16.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 2 (d) 4
  17. 17.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
  18. 18.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
  19. 19.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
  20. 20.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2219 (c) 2209 (d) 2809
  21. 21.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4891 (d) 4489
  22. 22.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 1235 (c) 925 (d) 625
  23. 23.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 144 (c) 74 (d) 4
  24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
  25. 25.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
  26. 26.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 9 (c) 7 (d) 8
  27. 27.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6396 (d) 6084
  29. 29.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 971 (d) 901
  31. 31.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 31 (d) 29
  32. 32.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  33. 33.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 7 (d) 6
  34. 34.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 433 (d) 289
  35. 35.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
  36. 36.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 529 (d) 409
  37. 37.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 39 (c) 13 (d) 41
  38. 38.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1575 (d) 1225
  39. 39.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 6425 (d) 7235
  40. 40.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 253 (d) 169

Answer key

  1. 10
  2. 576
  3. 17
  4. 164
  5. 10
  6. 202
  7. 12
  8. (x + 2)(x + 8)
  9. 841
  10. 5929
  11. 4624
  12. 11
  13. 625
  14. 63
  15. 5329
  16. 4
  17. 35
  18. 3025
  19. 9
  20. 2209
  21. 4489
  22. 1225
  23. 4
  24. 16
  25. 1444
  26. 8
  27. (x + 3)(x + 4)
  28. 6084
  29. (x + 6)(x + 7)
  30. 961
  31. 30
  32. (x + 7)(x + 8)
  33. 5
  34. 145
  35. 185
  36. 529
  37. 40
  38. 1225
  39. 7225
  40. 85

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