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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 60 · 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.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 100 (c) 196 (d) 148
  2. 2.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 3 and 8 (c) 1 and 24 (d) 2 and 12
  3. 3.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4916 (c) 5476 (d) 4356
  4. 4.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  6. 6.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
  7. 7.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
  8. 8.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 400 (d) 208
  9. 9.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 4904 (d) 5184
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
  11. 11.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 324 (d) 4
  12. 12.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 11 (d) 21
  13. 13.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 10 (d) 196
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 10609 (d) 9409
  16. 16.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 5939 (c) 6391 (d) 6889
  17. 17.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  18. 18.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
  19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
  20. 20.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 2 and 10 (c) 4 and 5 (d) 1 and 8
  21. 21.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 1 (d) 113
  22. 22.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 3 and 5 (d) 1 and 7
  23. 23.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 635 (d) 625
  24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
  25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  26. 26.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 43 (d) 41
  27. 27.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 4106 (d) 3136
  29. 29.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 29 (d) 31
  30. 30.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 24 (c) 11 (d) 10
  31. 31.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 13 (d) 12
  32. 32.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  33. 33.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
  34. 34.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 84 (c) 132 (d) 68
  35. 35.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1596 (d) 1764
  36. 36.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2491 (c) 2219 (d) 2809
  37. 37.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3584 (d) 3136
  38. 38.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
  39. 39.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 3 (d) 361
  40. 40.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 28 (d) 36

Answer key

  1. 100
  2. 3 and 8
  3. 5476
  4. 1681
  5. (x + 7)(x + 9)
  6. 16
  7. (x + 4)(x + 5)
  8. 208
  9. 5184
  10. 15
  11. 4
  12. 10
  13. 100
  14. (x + 7)(x + 8)
  15. 9409
  16. 5929
  17. 6 and 7
  18. 164
  19. (x + 3)(x + 7)
  20. 4 and 5
  21. 1
  22. 3 and 5
  23. 625
  24. 181
  25. (x + 3)(x + 5)
  26. 42
  27. (x + 3)(x + 8)
  28. 4096
  29. 30
  30. 11
  31. 13
  32. (x + 2)(x + 6)
  33. 13
  34. 68
  35. 1444
  36. 2209
  37. 3136
  38. (x + 5)(x + 6)
  39. 9
  40. 20

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