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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 324 · 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.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  2. 2.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 117 (d) 3
  3. 3.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6571 (c) 6241 (d) 6561
  4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
  5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1444 (d) 1764
  7. 7.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
  8. 8.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 9 (c) 3 (d) 169
  9. 9.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 23 (d) 25
  10. 10.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 25 (c) 23 (d) 11
  11. 11.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 2 and 18 (d) 1 and 12
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
  14. 14.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 7
  15. 15.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 213 (c) 147 (d) 169
  16. 16.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  17. 17.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 49 (d) 64
  18. 18.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 201 (d) 146
  19. 19.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 7 (c) 13 (d) 5
  20. 20.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 256 (d) 384
  21. 21.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 7744 (c) 8464 (d) 8096
  22. 22.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
  23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  25. 25.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 253 (d) 169
  26. 26.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 1 and 48 (d) 6 and 8
  27. 27.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
  28. 28.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)
  29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  30. 30.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 8 (c) 12 (d) 6
  31. 31.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 149 (d) 3
  32. 32.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 13 (d) 36
  33. 33.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 6384 (d) 5786
  34. 34.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 64 (d) 34
  35. 35.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  36. 36.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 36 (c) 2 (d) 20
  37. 37.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  38. 38.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  39. 39.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)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2809 (c) 2209 (d) 2819

Answer key

  1. (x + 4)(x + 5)
  2. 9
  3. 6561
  4. (x + 4)(x + 8)
  5. (x + 6)(x + 8)
  6. 1444
  7. (x + 5)(x + 7)
  8. 9
  9. 24
  10. 24
  11. 4 and 9
  12. (x + 2)(x + 8)
  13. 961
  14. 2 and 6
  15. 125
  16. 1936
  17. 34
  18. 146
  19. 5
  20. 256
  21. 7744
  22. 20
  23. 17
  24. (x + 3)(x + 6)
  25. 85
  26. 6 and 8
  27. 1
  28. (x + 5)(x + 8)
  29. (x + 8)(x + 9)
  30. 7
  31. 9
  32. 13
  33. 5776
  34. 4
  35. 63
  36. 4
  37. 361
  38. (x + 2)(x + 5)
  39. (x + 6)(x + 9)
  40. 2809

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