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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 203 · 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.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)
  2. 2.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 11 (c) 19 (d) 18
  3. 3.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 529 (d) 289
  4. 4.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  5. 5.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 13 (d) 1 and 48
  6. 6.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 196 (d) 100
  7. 7.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 7579 (c) 8649 (d) 8091
  8. 8.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1225 (c) 2035 (d) 1625
  9. 9.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 13 (d) 39
  10. 10.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 52
  11. 11.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)
  12. 12.If a + b = 12 and ab = 32, what is (a − b)²? (a) 4 (b) 80 (c) 144 (d) 16
  13. 13.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 15 (d) 8
  14. 14.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 9
  15. 15.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
  16. 16.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 11 (c) 10 (d) 16
  17. 17.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  18. 18.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 49 (d) 34
  19. 19.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 49 (d) 137
  20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  21. 21.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
  22. 22.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  23. 23.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 169 (c) 145 (d) 193
  24. 24.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  25. 25.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
  26. 26.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 62 (d) 64
  27. 27.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4624 (c) 4896 (d) 5184
  28. 28.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 5 and 8 (c) 2 and 20 (d) 1 and 12
  29. 29.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 40 (c) 16 (d) 64
  30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  31. 31.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 313 (d) 181
  32. 32.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 34 (c) 36 (d) 35
  33. 33.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28
  34. 34.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
  35. 35.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 181 (c) 121 (d) 91
  36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  37. 37.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 441
  38. 38.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
  39. 39.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
  40. 40.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 625 (c) 225 (d) 635

Answer key

  1. (x + 8)(x + 9)
  2. 18
  3. 289
  4. (x + 4)(x + 7)
  5. 6 and 8
  6. 100
  7. 7569
  8. 2025
  9. 40
  10. 4
  11. (x + 4)(x + 8)
  12. 16
  13. 8
  14. 8
  15. 361
  16. 10
  17. (x + 6)(x + 9)
  18. 34
  19. 49
  20. (x + 7)(x + 8)
  21. 16
  22. 20
  23. 145
  24. (x + 5)(x + 7)
  25. 6
  26. 63
  27. 4624
  28. 5 and 8
  29. 16
  30. (x + 6)(x + 8)
  31. 137
  32. 35
  33. 28
  34. (x + 3)(x + 4)
  35. 61
  36. 17
  37. 9
  38. 193
  39. 37
  40. 625

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