← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 998 · 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 44²? (a) 1616 (b) 1946 (c) 1936 (d) 1296
  2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10826 (d) 10816
  4. 4.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 1 and 13 (d) 3 and 15
  5. 5.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 49 (d) 3
  6. 6.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 37 (d) 25
  7. 7.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1681 (d) 1521
  8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
  10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  11. 11.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 8 (d) 6
  12. 12.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 1 and 6 (c) 2 and 3 (d) 2 and 4
  13. 13.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 101 (c) 121 (d) 141
  14. 14.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 225 (c) 117 (d) 171
  15. 15.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 441 (d) 225
  17. 17.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
  18. 18.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 2 (d) 256
  19. 19.What is the product of the roots of x² − 11x + 28 = 0? (a) 11 (b) 27 (c) 29 (d) 28
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  21. 21.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 5 (d) 81
  22. 22.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 100 (c) 122 (d) 144
  23. 23.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2209 (d) 2219
  24. 24.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
  25. 25.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 20 (d) 22
  26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 1 (d) 145
  27. 27.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 5625 (c) 7225 (d) 7235
  28. 28.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 382 (d) 256
  29. 29.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)
  30. 30.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 9 (c) 2 and 8 (d) 1 and 21
  31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  32. 32.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 256 (d) 586
  33. 33.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
  34. 34.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 4904 (d) 5184
  35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8659 (c) 8649 (d) 8109
  36. 36.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1531 (d) 1599
  37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10009 (d) 10619
  39. 39.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6401 (d) 6561
  40. 40.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 5 (d) 25

Answer key

  1. 1936
  2. (x + 7)(x + 9)
  3. 10816
  4. 5 and 9
  5. 9
  6. 25
  7. 1681
  8. (x + 5)(x + 6)
  9. (x + 3)(x + 4)
  10. (x + 5)(x + 8)
  11. 7
  12. 2 and 3
  13. 101
  14. 117
  15. 14
  16. 225
  17. (x + 3)(x + 7)
  18. 4
  19. 28
  20. (x + 8)(x + 9)
  21. 25
  22. 100
  23. 2209
  24. 8 and 9
  25. 21
  26. 1
  27. 7225
  28. 130
  29. (x + 6)(x + 8)
  30. 3 and 7
  31. 17
  32. 576
  33. 3025
  34. 5184
  35. 8649
  36. 1521
  37. (x + 4)(x + 5)
  38. 10609
  39. 6561
  40. 25

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