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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 132 · 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 82²? (a) 6724 (b) 6084 (c) 6734 (d) 6404
  2. 2.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)
  3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  4. 4.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  7. 7.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 1 and 42 (d) 6 and 7
  8. 8.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
  10. 10.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1521 (d) 1681
  11. 11.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3969 (d) 3609
  12. 12.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 104 (d) 124
  13. 13.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1946 (c) 1616 (d) 1296
  14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  15. 15.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
  16. 16.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 9
  17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 256 (d) 576
  18. 18.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3481 (c) 3721 (d) 3599
  19. 19.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
  20. 20.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 841 (d) 899
  21. 21.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
  22. 22.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 193 (c) 169 (d) 145
  23. 23.What are the roots of x² − 8x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 7 (d) 1 and 12
  24. 24.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361
  25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  26. 26.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 36 (c) 46 (d) 31
  27. 27.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
  28. 28.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 63 (d) 1 and 15
  29. 29.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 1225 (c) 635 (d) 875
  30. 30.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 271 (c) 361 (d) 1
  31. 31.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
  32. 32.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
  33. 33.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6724 (d) 6084
  34. 34.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 14 (c) 13 (d) 9
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
  36. 36.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 65 (d) 97
  37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 64 (d) 16
  38. 38.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 1 (c) 81 (d) 41
  39. 39.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625
  40. 40.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 6 (d) 180

Answer key

  1. 6724
  2. (x + 8)(x + 9)
  3. (x + 5)(x + 8)
  4. 149
  5. (x + 7)(x + 9)
  6. 16
  7. 6 and 7
  8. (x + 3)(x + 9)
  9. 1089
  10. 1521
  11. 3969
  12. 104
  13. 1936
  14. (x + 2)(x + 3)
  15. 5 and 7
  16. 8
  17. 576
  18. 3481
  19. 1444
  20. 841
  21. 48
  22. 145
  23. 2 and 6
  24. 181
  25. (x + 6)(x + 9)
  26. 26
  27. 3364
  28. 7 and 9
  29. 625
  30. 1
  31. 4
  32. 54
  33. 6084
  34. 14
  35. 15
  36. 65
  37. 63
  38. 1
  39. 5625
  40. 36

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