← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 566 · 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.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
  2. 2.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 9 (d) 7
  3. 3.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 6384 (c) 7056 (d) 5786
  5. 5.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  6. 6.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 65 (d) 121
  7. 7.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)
  8. 8.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 23 (d) 24
  9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2509 (d) 2819
  10. 10.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 33 (d) 12
  11. 11.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 37 (d) 36
  12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  13. 13.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
  14. 14.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  15. 15.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 1099 (d) 909
  16. 16.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1369 (d) 1859
  17. 17.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
  18. 18.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  19. 19.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 223 (d) 421
  20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  21. 21.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 875 (d) 635
  22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
  24. 24.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 11 (c) 8 (d) 12
  25. 25.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
  26. 26.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 1 and 32 (d) 4 and 8
  27. 27.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 225 (d) 3
  28. 28.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 7 (d) 15
  29. 29.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
  30. 30.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
  31. 31.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 100 (c) 68 (d) 84
  32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 925 (d) 625
  33. 33.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
  34. 34.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  35. 35.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10001 (c) 9801 (d) 10211
  36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  38. 38.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  39. 39.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 8 (d) 5
  40. 40.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31

Answer key

  1. 45
  2. 8
  3. (x + 6)(x + 9)
  4. 5776
  5. (x + 2)(x + 8)
  6. 65
  7. (x + 6)(x + 8)
  8. 24
  9. 2809
  10. 32
  11. 36
  12. 72
  13. 6 and 8
  14. 484
  15. 1089
  16. 1849
  17. (x + 3)(x + 7)
  18. 3 and 4
  19. 157
  20. (x + 7)(x + 9)
  21. 625
  22. (x + 3)(x + 5)
  23. 529
  24. 12
  25. 5184
  26. 4 and 8
  27. 9
  28. 8
  29. 202
  30. 4
  31. 68
  32. 1225
  33. 10
  34. (x + 5)(x + 9)
  35. 10201
  36. (x + 8)(x + 9)
  37. (x + 7)(x + 8)
  38. (x + 2)(x + 3)
  39. 6
  40. 30

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad