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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 50 · 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 + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  2. 2.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 361 (d) 181
  3. 3.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
  4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  5. 5.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
  6. 6.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
  7. 7.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)
  8. 8.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 73 (c) 81 (d) 65
  9. 9.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 2 and 3 (c) 2 and 4 (d) 1 and 4
  10. 10.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 10 (c) 1 and 28 (d) 2 and 14
  11. 11.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 324 (c) 16 (d) 4
  12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 16 (d) 14
  13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
  14. 14.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 5 (d) 157
  15. 15.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)
  16. 16.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 145 (c) 121 (d) 169
  17. 17.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
  18. 18.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
  19. 19.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 10 (d) 9
  20. 20.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 72 (d) 1 and 16
  21. 21.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
  22. 22.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 36 (d) 34
  23. 23.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)
  24. 24.If a + b = 9 and ab = 20, what is a² + b²? (a) 41 (b) 121 (c) 81 (d) 61
  25. 25.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 13 (c) 1 and 45 (d) 3 and 15
  26. 26.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
  27. 27.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 5 (d) 7
  28. 28.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3596 (d) 3364
  29. 29.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 27 (c) 28 (d) 29
  30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 14 (d) 54
  31. 31.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
  32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  33. 33.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 1 (c) 81 (d) 61
  34. 34.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
  35. 35.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
  36. 36.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8091 (d) 8649
  37. 37.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
  38. 38.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
  39. 39.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
  40. 40.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 5

Answer key

  1. (x + 5)(x + 7)
  2. 181
  3. 4
  4. (x + 6)(x + 7)
  5. 185
  6. 484
  7. (x + 2)(x + 5)
  8. 65
  9. 2 and 3
  10. 4 and 7
  11. 16
  12. 15
  13. 1089
  14. 25
  15. (x + 8)(x + 9)
  16. 121
  17. 5476
  18. (x + 2)(x + 4)
  19. 9
  20. 8 and 9
  21. 145
  22. 35
  23. (x + 5)(x + 9)
  24. 41
  25. 5 and 9
  26. 63
  27. 1
  28. 3364
  29. 28
  30. 15
  31. 4761
  32. (x + 5)(x + 8)
  33. 1
  34. 1156
  35. 961
  36. 7569
  37. (x + 3)(x + 4)
  38. 7
  39. (x + 2)(x + 9)
  40. 25

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