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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 936 · 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 sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
  2. 2.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
  3. 3.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 20 (d) 4
  4. 4.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 11 (d) 24
  5. 5.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 2 and 6 (c) 1 and 7 (d) 3 and 5
  6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9226 (d) 9984
  8. 8.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 289 (c) 5 (d) 25
  9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 7 (c) 8 (d) 6
  10. 10.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 14 (c) 9 (d) 10
  11. 11.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 273 (d) 537
  15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  16. 16.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10404 (d) 10414
  17. 17.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 8846 (c) 7396 (d) 8116
  18. 18.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 5 (d) 8
  19. 19.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2116 (d) 2916
  20. 20.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 21 (d) 9
  21. 21.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 376 (d) 136
  22. 22.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2509 (d) 2809
  24. 24.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 130 (d) 4
  25. 25.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 925 (d) 1235
  26. 26.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 8104 (c) 8474 (d) 7744
  27. 27.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 35 (d) 1 and 11
  28. 28.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
  29. 29.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 5041 (d) 4901
  30. 30.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 2 and 6 (c) 1 and 6 (d) 1 and 12
  31. 31.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)
  32. 32.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 55 (c) 15 (d) 53
  33. 33.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 40 (d) 41
  34. 34.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
  35. 35.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
  36. 36.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
  37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 11 (d) 12
  39. 39.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 441 (d) 371
  40. 40.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 6 (c) 100 (d) 68

Answer key

  1. 15
  2. (x + 2)(x + 5)
  3. 4
  4. 24
  5. 3 and 5
  6. (x + 5)(x + 9)
  7. 9216
  8. 25
  9. 7
  10. 9
  11. (x + 3)(x + 6)
  12. 576
  13. (x + 7)(x + 9)
  14. 185
  15. 145
  16. 10404
  17. 8836
  18. 6
  19. 2916
  20. 20
  21. 136
  22. (x + 2)(x + 8)
  23. 2809
  24. 4
  25. 1225
  26. 8464
  27. 5 and 7
  28. 4
  29. 5041
  30. 3 and 4
  31. (x + 6)(x + 7)
  32. 54
  33. 40
  34. 36
  35. (x + 3)(x + 8)
  36. 4
  37. 17
  38. 11
  39. 361
  40. 36

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