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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 692 · 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.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2714 (d) 2304
  3. 3.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 5 and 7 (d) 1 and 11
  4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
  6. 6.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
  7. 7.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 47 (c) 48 (d) 49
  8. 8.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1034 (d) 1024
  9. 9.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 17 (c) 19 (d) 9
  10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529
  12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 14 (d) 13
  13. 13.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 15 (d) 63
  14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  15. 15.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 202 (d) 2
  16. 16.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 198 (d) 117
  17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10001 (c) 10201 (d) 9801
  19. 19.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 1 and 54 (c) 2 and 27 (d) 6 and 9
  20. 20.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 3 (d) 9
  21. 21.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3844 (d) 3364
  23. 23.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2025 (d) 2525
  24. 24.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 9 (d) 25
  25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
  26. 26.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
  27. 27.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)
  28. 28.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
  29. 29.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
  30. 30.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 148 (d) 100
  31. 31.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 65 (c) 81 (d) 97
  32. 32.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
  33. 33.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  34. 34.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 10 (d) 9
  35. 35.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2219 (d) 2209
  36. 36.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 31 (d) 33
  37. 37.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 20 (d) 19
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  39. 39.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 10 (d) 9
  40. 40.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4499 (c) 5329 (d) 4489

Answer key

  1. (x + 2)(x + 9)
  2. 2704
  3. 5 and 7
  4. (x + 7)(x + 9)
  5. 6724
  6. 8836
  7. 48
  8. 1024
  9. 18
  10. (x + 8)(x + 9)
  11. 289
  12. 14
  13. 16
  14. (x + 5)(x + 6)
  15. 4
  16. 90
  17. (x + 6)(x + 7)
  18. 10201
  19. 6 and 9
  20. 9
  21. (x + 3)(x + 6)
  22. 3364
  23. 3025
  24. 9
  25. 1936
  26. 2 and 5
  27. (x + 7)(x + 8)
  28. 56
  29. 10816
  30. 100
  31. 65
  32. 9
  33. (x + 3)(x + 9)
  34. 9
  35. 2209
  36. 32
  37. 20
  38. 17
  39. 10
  40. 4489

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