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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 929 · 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 + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  2. 2.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 45 (d) 14
  3. 3.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  4. 4.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 1 and 6 (c) 2 and 6 (d) 3 and 4
  5. 5.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  6. 6.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
  7. 7.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 333 (d) 117
  8. 8.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 1 and 10 (d) 2 and 9
  9. 9.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
  10. 10.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 225 (d) 625
  11. 11.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
  12. 12.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3025 (c) 3575 (d) 3035
  13. 13.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 202 (d) 4
  14. 14.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 324 (d) 4
  15. 15.If a + b = 12 and ab = 27, what is (a − b)²? (a) 144 (b) 36 (c) 90 (d) 6
  16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 106 (d) 286
  17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1369 (d) 1379
  20. 20.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6241 (d) 6401
  21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  22. 22.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 1 and 11 (c) 4 and 8 (d) 2 and 16
  23. 23.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
  24. 24.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 625 (d) 1225
  25. 25.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 409 (d) 289
  26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  27. 27.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)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 11025 (d) 9025
  29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  30. 30.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 12 (c) 4 and 9 (d) 1 and 36
  31. 31.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 541 (d) 361
  32. 32.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10404 (c) 10414 (d) 10004
  33. 33.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 16 (d) 14
  34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  35. 35.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)
  36. 36.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 89 (d) 169
  37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
  38. 38.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 4 (d) 2
  39. 39.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 62 (d) 64
  40. 40.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6251 (d) 6241

Answer key

  1. (x + 2)(x + 3)
  2. 14
  3. (x + 2)(x + 4)
  4. 3 and 4
  5. 17
  6. 2 and 4
  7. 117
  8. 2 and 9
  9. 42
  10. 625
  11. 37
  12. 3025
  13. 4
  14. 16
  15. 36
  16. 106
  17. 14
  18. (x + 7)(x + 8)
  19. 1369
  20. 6561
  21. (x + 4)(x + 7)
  22. 4 and 8
  23. 4
  24. 625
  25. 529
  26. 15
  27. (x + 6)(x + 9)
  28. 11025
  29. (x + 6)(x + 8)
  30. 4 and 9
  31. 181
  32. 10404
  33. 15
  34. (x + 8)(x + 9)
  35. (x + 7)(x + 9)
  36. 89
  37. 441
  38. 4
  39. 63
  40. 6241

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