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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 831 · 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.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
  2. 2.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 9 (c) 1 and 21 (d) 3 and 7
  3. 3.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
  4. 4.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 196 (c) 148 (d) 100
  5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  6. 6.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
  7. 7.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 13 (d) 42
  8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 625 (d) 1235
  11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x − 5)(x − 6) (d) (x + 1)(x + 10)
  12. 12.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 109 (c) 139 (d) 169
  13. 13.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  14. 14.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
  16. 16.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
  17. 17.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 1 and 11 (d) 2 and 10
  18. 18.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 6 (d) 64
  19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  20. 20.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 8 (c) 9 (d) 12
  21. 21.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
  24. 24.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 4904 (d) 5184
  25. 25.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10201 (d) 10211
  26. 26.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 25 (c) 5 (d) 37
  27. 27.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9216 (d) 9984
  28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  29. 29.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
  30. 30.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  31. 31.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 225 (c) 625 (d) 375
  32. 32.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3601 (d) 3731
  33. 33.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 6 (c) 144 (d) 90
  34. 34.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 11 (c) 25 (d) 24
  35. 35.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 529 (d) 289
  36. 36.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 56 (d) 15
  37. 37.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  38. 38.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235
  39. 39.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
  40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 164 (d) 4

Answer key

  1. 4
  2. 3 and 7
  3. (x + 4)(x + 8)
  4. 100
  5. (x + 8)(x + 9)
  6. 130
  7. 42
  8. (x + 3)(x + 7)
  9. (x + 5)(x + 8)
  10. 1225
  11. (x + 5)(x + 6)
  12. 109
  13. 145
  14. (x + 3)(x + 8)
  15. 2601
  16. 10
  17. 3 and 9
  18. 36
  19. (x + 4)(x + 7)
  20. 8
  21. (x + 4)(x + 9)
  22. (x + 7)(x + 9)
  23. 181
  24. 5184
  25. 10201
  26. 25
  27. 9216
  28. 17
  29. 1
  30. (x + 6)(x + 9)
  31. 225
  32. 3721
  33. 36
  34. 24
  35. 289
  36. 56
  37. 36
  38. 1225
  39. 13
  40. 4

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