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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 941 · 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 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
  3. 3.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 63 (d) 81
  4. 4.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13
  5. 5.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 5 (d) 6
  6. 6.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11035 (d) 11025
  7. 7.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 7 (c) 6 (d) 8
  10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  11. 11.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
  12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  13. 13.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 37 (c) 25 (d) 1
  14. 14.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
  15. 15.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 333 (c) 117 (d) 171
  16. 16.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9999 (d) 9811
  17. 17.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10001 (d) 10201
  18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1
  19. 19.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 109 (c) 169 (d) 229
  20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  21. 21.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
  22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  23. 23.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 2 (d) 144
  24. 24.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 7754 (d) 8464
  25. 25.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  27. 27.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 252 (d) 180
  28. 28.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 14 (d) 47
  29. 29.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
  30. 30.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1936 (d) 1584
  32. 32.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 1 and 24 (d) 3 and 8
  33. 33.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 400 (d) 2
  34. 34.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
  35. 35.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 193 (d) 529
  36. 36.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
  37. 37.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  38. 38.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
  39. 39.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 10 (d) 4
  40. 40.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8

Answer key

  1. 7396
  2. (x + 4)(x + 5)
  3. 45
  4. 6 and 8
  5. 6
  6. 11025
  7. (x + 2)(x + 9)
  8. (x + 6)(x + 7)
  9. 7
  10. (x + 7)(x + 8)
  11. 63
  12. (x + 5)(x + 7)
  13. 1
  14. 49
  15. 117
  16. 9801
  17. 10201
  18. 1
  19. 109
  20. (x + 2)(x + 4)
  21. 6 and 9
  22. (x + 3)(x + 4)
  23. 4
  24. 7744
  25. (x + 2)(x + 6)
  26. (x + 8)(x + 9)
  27. 180
  28. 48
  29. 6561
  30. (x + 5)(x + 8)
  31. 1296
  32. 3 and 8
  33. 4
  34. 7 and 8
  35. 193
  36. 16
  37. (x + 2)(x + 3)
  38. 164
  39. 4
  40. 2 and 4

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