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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 894 · 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.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 9614 (d) 10404
  2. 2.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1306 (d) 1584
  3. 3.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 5 (d) 361
  4. 4.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 136 (c) 256 (d) 16
  5. 5.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 3 and 21 (d) 1 and 63
  6. 6.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
  7. 7.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 100 (d) 132
  8. 8.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 11 (d) 21
  9. 9.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 85 (d) 49
  10. 10.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 196 (d) 10
  11. 11.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 361 (d) 441
  12. 12.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 10
  13. 13.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 16 (d) 62
  14. 14.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
  15. 15.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 49 (c) 169 (d) 109
  16. 16.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 635 (d) 625
  17. 17.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611
  18. 18.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 45 (d) 14
  19. 19.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 1 and 20 (d) 4 and 5
  20. 20.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 289 (d) 299
  21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  22. 22.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 17 (d) 3
  23. 23.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
  24. 24.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7921 (c) 7931 (d) 8099
  25. 25.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 12 (c) 40 (d) 14
  26. 26.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 117 (d) 171
  27. 27.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3969 (d) 3249
  28. 28.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
  30. 30.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8104 (c) 7744 (d) 8464
  31. 31.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 10201 (d) 9999
  33. 33.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 8096 (d) 7754
  34. 34.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 1 and 6 (c) 2 and 5 (d) 3 and 4
  35. 35.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 24 (c) 2 and 12 (d) 1 and 10
  36. 36.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 97 (c) 73 (d) 121
  37. 37.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 10
  38. 38.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 52
  39. 39.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)

Answer key

  1. 9604
  2. 1296
  3. 25
  4. 16
  5. 7 and 9
  6. (x + 3)(x + 9)
  7. 68
  8. 10
  9. 49
  10. 100
  11. 441
  12. 2 and 9
  13. 63
  14. 170
  15. 49
  16. 625
  17. 2601
  18. 45
  19. 4 and 5
  20. 289
  21. (x + 2)(x + 3)
  22. 9
  23. 9
  24. 7921
  25. 13
  26. 117
  27. 3969
  28. (x + 2)(x + 8)
  29. 3721
  30. 8464
  31. (x + 5)(x + 8)
  32. 9801
  33. 7744
  34. 2 and 5
  35. 3 and 8
  36. 73
  37. 24
  38. 4
  39. 25
  40. (x + 6)(x + 7)

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