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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 355 · 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 − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
  2. 2.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  3. 3.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 409 (d) 289
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  5. 5.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  6. 6.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 97 (c) 133 (d) 169
  7. 7.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 6416 (d) 7056
  8. 8.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
  9. 9.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)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
  11. 11.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
  12. 12.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2525 (d) 2025
  13. 13.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  14. 14.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 36 (d) 6
  15. 15.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 9 (c) 3 (d) 289
  16. 16.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 49 (c) 25 (d) 73
  17. 17.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
  19. 19.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 416 (d) 586
  20. 20.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 136 (c) 4 (d) 16
  21. 21.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 37 (c) 25 (d) 49
  22. 22.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 169 (d) 85
  23. 23.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 112 (c) 208 (d) 80
  24. 24.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 21 (d) 10
  25. 25.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 26 (c) 28 (d) 12
  26. 26.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 4225 (d) 3575
  27. 27.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 4 and 5 (c) 1 and 20 (d) 2 and 10
  28. 28.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 202 (d) 400
  29. 29.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)
  30. 30.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 37 (d) 13
  31. 31.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  32. 32.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  33. 33.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
  34. 34.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 11 (d) 30
  35. 35.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
  36. 36.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 61 (d) 81
  37. 37.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 32 (d) 11
  38. 38.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3969 (d) 3249
  39. 39.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 11 (c) 1 and 32 (d) 2 and 16
  40. 40.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)

Answer key

  1. 54
  2. (x + 3)(x + 8)
  3. 169
  4. (x + 8)(x + 9)
  5. (x + 6)(x + 7)
  6. 97
  7. 7056
  8. (x + 2)(x + 9)
  9. (x + 7)(x + 8)
  10. 6724
  11. 11
  12. 3025
  13. 17
  14. 36
  15. 9
  16. 25
  17. (x + 5)(x + 6)
  18. 5329
  19. 576
  20. 16
  21. 25
  22. 1
  23. 80
  24. 21
  25. 27
  26. 3025
  27. 4 and 5
  28. 202
  29. (x + 3)(x + 4)
  30. 13
  31. 15
  32. 1681
  33. 17
  34. 30
  35. 170
  36. 1
  37. 12
  38. 3249
  39. 4 and 8
  40. (x + 5)(x + 8)

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