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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 29 · 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 39²? (a) 1599 (b) 1521 (c) 1531 (d) 1681
  2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101
  3. 3.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 52 (d) 88
  4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  5. 5.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 404 (c) 484 (d) 324
  7. 7.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  8. 8.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 169 (d) 25
  9. 9.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 127 (d) 1
  10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 904 (c) 1034 (d) 784
  12. 12.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
  13. 13.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3601 (d) 3731
  14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 361 (d) 181
  15. 15.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 13 (d) 19
  16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  17. 17.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
  18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 145 (d) 217
  19. 19.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 325 (d) 175
  20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  22. 22.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 8 (c) 1 and 5 (d) 2 and 5
  23. 23.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
  24. 24.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  26. 26.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 42 (d) 14
  27. 27.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 256 (d) 576
  28. 28.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 144 (c) 184 (d) 124
  29. 29.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 160 (c) 64 (d) 256
  30. 30.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
  31. 31.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 52 (d) 2
  32. 32.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
  34. 34.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 130 (c) 196 (d) 8
  35. 35.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 121 (c) 9 (d) 65
  36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  37. 37.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 145 (d) 433
  38. 38.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 117 (c) 81 (d) 45
  39. 39.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10816 (d) 10826
  40. 40.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 81 (d) 121

Answer key

  1. 1521
  2. 8281
  3. 40
  4. (x + 5)(x + 8)
  5. (x + 5)(x + 6)
  6. 484
  7. (x + 2)(x + 4)
  8. 25
  9. 1
  10. 15
  11. 1024
  12. 4
  13. 3721
  14. 1
  15. 13
  16. (x + 6)(x + 9)
  17. 4
  18. 1
  19. 125
  20. (x + 7)(x + 9)
  21. (x + 4)(x + 5)
  22. 2 and 4
  23. 9
  24. (x + 2)(x + 3)
  25. (x + 6)(x + 8)
  26. 13
  27. 256
  28. 104
  29. 64
  30. (x + 2)(x + 9)
  31. 4
  32. (x + 7)(x + 8)
  33. 6724
  34. 64
  35. 9
  36. (x + 8)(x + 9)
  37. 145
  38. 45
  39. 10816
  40. 41

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