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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 932 · 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.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  2. 2.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  4. 4.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 451 (d) 401
  5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6561 (d) 6241
  6. 6.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 11 (d) 29
  7. 7.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 539 (c) 529 (d) 289
  8. 8.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 289 (c) 299 (d) 529
  9. 9.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  11. 11.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 244 (c) 196 (d) 172
  12. 12.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 2 and 21 (d) 1 and 42
  13. 13.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
  14. 14.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 11 (c) 10 (d) 21
  15. 15.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
  16. 16.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1591 (d) 1379
  17. 17.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 16 (c) 4 (d) 26
  18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  19. 19.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
  20. 20.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3481 (d) 3491
  21. 21.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 169 (c) 129 (d) 249
  22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9801 (d) 9999
  23. 23.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)
  24. 24.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 252 (d) 324
  25. 25.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
  26. 26.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  27. 27.What are the roots of x² − 12x + 27 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 27 (d) 3 and 9
  28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9991 (c) 10609 (d) 9419
  30. 30.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 193 (c) 169 (d) 157
  31. 31.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  32. 32.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 100 (c) 58 (d) 16
  33. 33.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 36 (d) 116
  34. 34.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  35. 35.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  36. 36.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 1 and 10 (d) 5 and 6
  37. 37.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
  38. 38.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
  39. 39.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 13 (d) 19
  40. 40.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56

Answer key

  1. 17
  2. 8
  3. (x + 8)(x + 9)
  4. 441
  5. 6241
  6. 30
  7. 529
  8. 289
  9. (x + 3)(x + 9)
  10. (x + 7)(x + 9)
  11. 148
  12. 6 and 7
  13. (x + 2)(x + 9)
  14. 10
  15. 1
  16. 1369
  17. 16
  18. (x + 2)(x + 3)
  19. 208
  20. 3481
  21. 89
  22. 9801
  23. (x + 5)(x + 6)
  24. 180
  25. 25
  26. (x + 2)(x + 5)
  27. 3 and 9
  28. (x + 7)(x + 8)
  29. 9409
  30. 145
  31. 14
  32. 16
  33. 36
  34. (x + 2)(x + 6)
  35. (x + 6)(x + 8)
  36. 5 and 6
  37. 130
  38. 1024
  39. 13
  40. 56

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