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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 272 · 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 17²? (a) 391 (b) 289 (c) 299 (d) 529
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  3. 3.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 1 and 8 (c) 2 and 4 (d) 2 and 5
  4. 4.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9035 (d) 9025
  5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  7. 7.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
  9. 9.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 104 (d) 144
  10. 10.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)
  11. 11.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 10 (c) 12 (d) 18
  12. 12.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 13 (c) 39 (d) 40
  13. 13.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 9 (c) 11 (d) 7
  14. 14.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
  15. 15.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2916 (d) 2116
  16. 16.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 289 (d) 421
  17. 17.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 7 (c) 2 and 6 (d) 1 and 15
  18. 18.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 31 (d) 30
  19. 19.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3481 (d) 3731
  20. 20.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  21. 21.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 3025 (d) 2525
  22. 22.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 1 and 18 (c) 2 and 9 (d) 3 and 6
  23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  24. 24.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  25. 25.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 2 and 27 (d) 6 and 9
  26. 26.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  27. 27.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10826 (d) 10016
  29. 29.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5329 (c) 5339 (d) 4909
  30. 30.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 12 (c) 40 (d) 14
  31. 31.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 9801 (d) 10201
  32. 32.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  33. 33.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
  34. 34.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2209 (c) 2809 (d) 2219
  35. 35.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 361 (d) 193
  36. 36.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
  37. 37.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  39. 39.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 1 and 10 (c) 4 and 7 (d) 1 and 28
  40. 40.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)

Answer key

  1. 289
  2. 17
  3. 2 and 4
  4. 9025
  5. (x + 6)(x + 8)
  6. (x + 5)(x + 6)
  7. (x + 4)(x + 9)
  8. 7569
  9. 104
  10. (x + 7)(x + 8)
  11. 11
  12. 40
  13. 10
  14. 164
  15. 2916
  16. 157
  17. 3 and 5
  18. 30
  19. 3721
  20. 48
  21. 3025
  22. 3 and 6
  23. (x + 7)(x + 9)
  24. 72
  25. 6 and 9
  26. 3 and 4
  27. (x + 3)(x + 6)
  28. 10816
  29. 5329
  30. 13
  31. 9801
  32. (x + 8)(x + 9)
  33. 9
  34. 2209
  35. 25
  36. 9
  37. (x + 2)(x + 3)
  38. 16
  39. 4 and 7
  40. (x + 4)(x + 6)

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