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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 262 · 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 + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  5. 5.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 54 (c) 6 and 9 (d) 1 and 14
  6. 6.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 12 (d) 30
  7. 7.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  8. 8.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 40 (c) 64 (d) 52
  9. 9.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 5 (d) 8
  10. 10.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 45 (d) 14
  11. 11.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
  12. 12.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 225 (d) 325
  13. 13.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
  14. 14.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 289 (d) 5
  15. 15.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
  16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  17. 17.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
  18. 18.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 1 and 8 (d) 2 and 4
  19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  20. 20.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  21. 21.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  22. 22.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)
  23. 23.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 9 (c) 13 (d) 5
  24. 24.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  25. 25.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 180 (c) 324 (d) 6
  26. 26.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  27. 27.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 196 (d) 100
  28. 28.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 169 (c) 125 (d) 81
  29. 29.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)
  30. 30.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 136 (d) 256
  31. 31.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
  32. 32.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 9 (c) 19 (d) 18
  33. 33.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  34. 34.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 144 (c) 10 (d) 122
  35. 35.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 15 (d) 8
  36. 36.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
  37. 37.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 3025 (c) 2475 (d) 2035
  38. 38.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 1 (c) 19 (d) 25
  39. 39.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 2 and 28 (d) 1 and 14
  40. 40.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333

Answer key

  1. (x + 5)(x + 8)
  2. 17
  3. (x + 5)(x + 9)
  4. (x + 8)(x + 9)
  5. 6 and 9
  6. 11
  7. 56
  8. 40
  9. 6
  10. 45
  11. 42
  12. 125
  13. (x + 3)(x + 8)
  14. 25
  15. 484
  16. (x + 4)(x + 9)
  17. 181
  18. 2 and 4
  19. 15
  20. (x + 6)(x + 8)
  21. (x + 4)(x + 8)
  22. (x + 3)(x + 4)
  23. 5
  24. 1024
  25. 36
  26. 208
  27. 4
  28. 81
  29. (x + 6)(x + 7)
  30. 136
  31. 8 and 9
  32. 18
  33. 6 and 7
  34. 100
  35. 15
  36. 529
  37. 2025
  38. 1
  39. 7 and 8
  40. 225

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