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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 719 · 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 91²? (a) 8101 (b) 7921 (c) 8281 (d) 8291
  2. 2.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 48 (d) 13
  3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 15 (d) 17
  4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1089 (d) 1099
  6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  7. 7.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  8. 8.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
  9. 9.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  10. 10.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)
  11. 11.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 9 (d) 8
  12. 12.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 10 (c) 2 (d) 4
  13. 13.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 18 (d) 10
  14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  15. 15.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 289 (c) 149 (d) 429
  16. 16.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 635 (d) 625
  17. 17.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 256 (c) 2 (d) 4
  18. 18.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 196 (c) 136 (d) 376
  19. 19.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
  21. 21.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 121 (c) 5 (d) 73
  22. 22.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  23. 23.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 375 (d) 225
  24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
  25. 25.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 16 (c) 15 (d) 14
  26. 26.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 371 (d) 441
  27. 27.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
  28. 28.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 5329 (d) 4499
  29. 29.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
  30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  31. 31.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 10 (c) 13 (d) 16
  32. 32.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 21 (c) 33 (d) 17
  33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  34. 34.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 6 (c) 2 and 4 (d) 1 and 4
  35. 35.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
  36. 36.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 169 (c) 125 (d) 81
  37. 37.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  39. 39.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 49 (d) 14
  40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)

Answer key

  1. 8281
  2. 14
  3. 16
  4. (x + 5)(x + 7)
  5. 1089
  6. (x + 4)(x + 6)
  7. (x + 4)(x + 9)
  8. 63
  9. (x + 4)(x + 7)
  10. (x + 3)(x + 6)
  11. 8
  12. 4
  13. 11
  14. (x + 6)(x + 7)
  15. 149
  16. 625
  17. 4
  18. 136
  19. 3249
  20. 145
  21. 25
  22. 164
  23. 225
  24. (x + 2)(x + 3)
  25. 15
  26. 361
  27. 9025
  28. 4489
  29. 6
  30. 17
  31. 10
  32. 17
  33. (x + 4)(x + 5)
  34. 2 and 3
  35. 56
  36. 81
  37. (x + 2)(x + 9)
  38. (x + 8)(x + 9)
  39. 48
  40. (x + 2)(x + 5)

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