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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 702 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  2. 2.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 10 (c) 6 (d) 7
  3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
  4. 4.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 225 (d) 635
  5. 5.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  6. 6.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  7. 7.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
  8. 8.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
  9. 9.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 4 (d) 16
  10. 10.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 1 and 7 (d) 3 and 5
  11. 11.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 19 (c) 13 (d) 37
  12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  13. 13.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 180 (d) 6
  14. 14.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 144 (d) 214
  15. 15.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 36 (d) 144
  16. 16.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 9 (d) 18
  17. 17.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 625 (d) 925
  18. 18.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 4 (c) 16 (d) 58
  19. 19.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
  20. 20.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 3025 (d) 2525
  21. 21.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 25 (c) 169 (d) 97
  22. 22.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 19 (c) 9 (d) 21
  23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  24. 24.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
  25. 25.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 6 (c) 3 and 4 (d) 1 and 10
  26. 26.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 225 (c) 375 (d) 625
  27. 27.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
  28. 28.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 164 (d) 2
  29. 29.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 8 (c) 15 (d) 7
  30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
  31. 31.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)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
  33. 33.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 2 and 10 (d) 4 and 5
  34. 34.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 441
  35. 35.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 69 (d) 39
  36. 36.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
  38. 38.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 142 (c) 58 (d) 79
  39. 39.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
  40. 40.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 39 (d) 41

Answer key

  1. (x + 8)(x + 9)
  2. 7
  3. 10201
  4. 625
  5. (x + 2)(x + 8)
  6. 1
  7. (x + 4)(x + 9)
  8. 113
  9. 16
  10. 3 and 5
  11. 13
  12. (x + 6)(x + 7)
  13. 36
  14. 74
  15. 36
  16. 9
  17. 1225
  18. 16
  19. 85
  20. 3025
  21. 25
  22. 20
  23. 17
  24. 10
  25. 2 and 5
  26. 225
  27. 25
  28. 4
  29. 8
  30. (x + 2)(x + 5)
  31. (x + 5)(x + 8)
  32. 7396
  33. 4 and 5
  34. 9
  35. 29
  36. (x + 6)(x + 8)
  37. 1444
  38. 58
  39. 6 and 7
  40. 40

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