← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 67 · 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 91²? (a) 8101 (b) 8281 (c) 8291 (d) 7921
  2. 2.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 2 (d) 64
  3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10201 (d) 10001
  4. 4.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 361 (c) 3 (d) 185
  5. 5.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  6. 6.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 11 (d) 9
  7. 7.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1369 (d) 1379
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1609 (c) 1859 (d) 1369
  10. 10.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2504 (d) 2714
  11. 11.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  12. 12.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 109 (d) 144
  13. 13.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 30 (d) 10
  14. 14.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 49 (c) 29 (d) 9
  15. 15.If a + b = 14 and ab = 48, what is a² + b²? (a) 100 (b) 196 (c) 148 (d) 292
  16. 16.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 169 (d) 157
  17. 17.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 7056 (c) 6384 (d) 5786
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 5041 (d) 4901
  20. 20.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4875 (c) 4235 (d) 5625
  21. 21.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4916 (c) 4356 (d) 5486
  22. 22.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)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5339 (c) 4489 (d) 5329
  24. 24.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  25. 25.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 784 (d) 896
  26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 6 (d) 8
  27. 27.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
  28. 28.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)
  29. 29.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 16 (c) 2 (d) 10
  30. 30.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 2 and 6 (d) 3 and 5
  31. 31.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
  32. 32.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
  33. 33.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 144 (c) 80 (d) 16
  34. 34.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 184 (d) 104
  35. 35.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 16 (d) 10
  36. 36.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
  37. 37.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 48 (d) 47
  38. 38.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  39. 39.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  40. 40.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 784 (c) 1024 (d) 904

Answer key

  1. 8281
  2. 4
  3. 10201
  4. 9
  5. (x + 3)(x + 5)
  6. 10
  7. 1369
  8. (x + 6)(x + 7)
  9. 1849
  10. 2704
  11. (x + 3)(x + 9)
  12. 74
  13. 11
  14. 9
  15. 100
  16. 145
  17. 5776
  18. (x + 2)(x + 8)
  19. 5041
  20. 4225
  21. 5476
  22. (x + 8)(x + 9)
  23. 5329
  24. 72
  25. 784
  26. 7
  27. 625
  28. (x + 7)(x + 8)
  29. 4
  30. 3 and 5
  31. 180
  32. 208
  33. 16
  34. 104
  35. 10
  36. (x + 5)(x + 9)
  37. 48
  38. 256
  39. (x + 6)(x + 8)
  40. 1024

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