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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 389 · 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) 541 (b) 361 (c) 181 (d) 271
  2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  4. 4.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  5. 5.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
  6. 6.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  8. 8.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 8 (c) 12 (d) 13
  9. 9.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 17 (c) 16 (d) 10
  10. 10.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3374 (d) 3844
  11. 11.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68
  12. 12.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 9604 (d) 10404
  13. 13.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
  14. 14.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 5 (c) 7 (d) 9
  15. 15.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
  16. 16.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
  17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  18. 18.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)
  19. 19.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  20. 20.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
  21. 21.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 100 (c) 68 (d) 6
  22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
  23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
  24. 24.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 81 (c) 53 (d) 109
  25. 25.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 91 (d) 121
  26. 26.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 1 and 8 (c) 2 and 9 (d) 3 and 6
  27. 27.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 19 (c) 1 (d) 13
  28. 28.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
  29. 29.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 61 (c) 49 (d) 43
  30. 30.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  31. 31.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 16 (d) 15
  32. 32.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 4 (d) 324
  33. 33.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 80 (c) 208 (d) 112
  34. 34.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8464 (d) 8096
  35. 35.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 8075 (d) 7225
  36. 36.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 8 (d) 7
  37. 37.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 5 and 8 (c) 2 and 20 (d) 1 and 40
  38. 38.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 34 (d) 35
  39. 39.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 400 (d) 202
  40. 40.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 8 (d) 7

Answer key

  1. 181
  2. (x + 6)(x + 9)
  3. (x + 8)(x + 9)
  4. 72
  5. 961
  6. 3721
  7. (x + 7)(x + 9)
  8. 12
  9. 16
  10. 3364
  11. 68
  12. 10404
  13. 1296
  14. 5
  15. 170
  16. 6 and 8
  17. (x + 6)(x + 8)
  18. (x + 5)(x + 8)
  19. 6 and 7
  20. 164
  21. 36
  22. 441
  23. 17
  24. 53
  25. 61
  26. 3 and 6
  27. 1
  28. 49
  29. 37
  30. (x + 6)(x + 7)
  31. 15
  32. 4
  33. 80
  34. 7744
  35. 7225
  36. 8
  37. 5 and 8
  38. 35
  39. 202
  40. 6

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