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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 710 · 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 = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
  2. 2.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3259 (c) 3969 (d) 3249
  3. 3.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 10 (d) 3 and 8
  4. 4.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 7 (c) 9 (d) 6
  5. 5.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 7 (c) 8 (d) 12
  6. 6.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1601 (d) 1691
  7. 7.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
  8. 8.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 16 (d) 100
  9. 9.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 56 (d) 15
  10. 10.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 25 (c) 53 (d) 81
  11. 11.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
  12. 12.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  13. 13.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 271 (d) 361
  14. 14.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  15. 15.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
  16. 16.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3481 (c) 3721 (d) 3599
  17. 17.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 43 (d) 13
  18. 18.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 5476 (c) 4916 (d) 5486
  19. 19.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
  20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 409 (c) 289 (d) 539
  23. 23.If a + b = 5 and ab = 4, what is a² + b²? (a) 33 (b) 21 (c) 17 (d) 25
  24. 24.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 196 (c) 106 (d) 4
  25. 25.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 185 (d) 9
  26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
  27. 27.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5635 (c) 7225 (d) 5625
  28. 28.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 37 (d) 35
  29. 29.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  30. 30.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  33. 33.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 124 (d) 104
  34. 34.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 91 (d) 1
  35. 35.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 4925 (c) 4225 (d) 5625
  36. 36.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 14 (d) 10
  37. 37.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
  38. 38.If a + b = 13 and ab = 12, what is (a − b)²? (a) 121 (b) 145 (c) 11 (d) 169
  39. 39.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4896 (d) 4624
  40. 40.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63

Answer key

  1. 181
  2. 3249
  3. 3 and 8
  4. 8
  5. 8
  6. 1681
  7. 24
  8. 16
  9. 15
  10. 25
  11. 5 and 6
  12. (x + 3)(x + 4)
  13. 1
  14. (x + 7)(x + 9)
  15. 36
  16. 3481
  17. 42
  18. 5476
  19. 36
  20. (x + 4)(x + 5)
  21. (x + 2)(x + 3)
  22. 529
  23. 17
  24. 16
  25. 9
  26. 7056
  27. 5625
  28. 36
  29. 2 and 3
  30. (x + 7)(x + 8)
  31. 361
  32. (x + 6)(x + 9)
  33. 104
  34. 1
  35. 5625
  36. 9
  37. 10201
  38. 121
  39. 4624
  40. 16

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