← Identities, polynomials and quadratic roots lesson New set →

Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 98 · 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 63²? (a) 3979 (b) 3249 (c) 3609 (d) 3969
  2. 2.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 164 (c) 2 (d) 4
  3. 3.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 118 (c) 91 (d) 82
  4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  6. 6.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 12 (d) 13
  7. 7.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3481 (c) 3721 (d) 3599
  8. 8.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10004 (c) 9604 (d) 10414
  9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  10. 10.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
  12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7396 (d) 7406
  14. 14.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 529 (d) 193
  15. 15.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  16. 16.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 163 (d) 262
  17. 17.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
  18. 18.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 10 (c) 4 (d) 2
  19. 19.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 73 (d) 37
  20. 20.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  21. 21.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 10 (c) 2 and 15 (d) 1 and 30
  22. 22.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
  23. 23.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 7 (c) 85 (d) 49
  24. 24.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 58 (c) 4 (d) 16
  25. 25.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10201 (c) 10211 (d) 10001
  26. 26.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 58 (c) 142 (d) 79
  27. 27.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
  28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
  29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
  30. 30.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 46 (d) 36
  31. 31.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 7 (d) 5
  32. 32.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 1 (c) 25 (d) 19
  33. 33.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  34. 34.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
  35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  36. 36.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 36 (c) 144 (d) 90
  37. 37.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11025 (d) 11035
  38. 38.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 64 (c) 8 (d) 130
  39. 39.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
  40. 40.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)

Answer key

  1. 3969
  2. 4
  3. 82
  4. (x + 6)(x + 7)
  5. (x + 4)(x + 5)
  6. 12
  7. 3481
  8. 10404
  9. (x + 5)(x + 8)
  10. (x + 2)(x + 7)
  11. 4624
  12. (x + 6)(x + 9)
  13. 7396
  14. 193
  15. (x + 8)(x + 9)
  16. 130
  17. 2116
  18. 4
  19. 25
  20. (x + 4)(x + 8)
  21. 5 and 6
  22. 185
  23. 49
  24. 16
  25. 10201
  26. 58
  27. 16
  28. 484
  29. 361
  30. 26
  31. 5
  32. 1
  33. (x + 6)(x + 8)
  34. (x + 3)(x + 6)
  35. (x + 7)(x + 9)
  36. 36
  37. 11025
  38. 64
  39. 2704
  40. (x + 3)(x + 4)

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