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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 447 · 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 are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 2 and 16 (c) 1 and 11 (d) 4 and 8
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  3. 3.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4624 (d) 4896
  4. 4.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 61 (d) 121
  5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  6. 6.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 23 (d) 25
  7. 7.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1531 (d) 1681
  8. 8.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6571 (c) 6401 (d) 6241
  9. 9.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 49 (c) 73 (d) 25
  10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  11. 11.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)
  12. 12.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
  13. 13.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  14. 14.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208
  15. 15.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 17 (c) 9 (d) 19
  16. 16.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6399 (d) 6251
  17. 17.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 256 (d) 4
  18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  19. 19.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 181 (d) 361
  20. 20.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 7 (d) 4
  21. 21.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
  22. 22.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 181 (c) 61 (d) 121
  23. 23.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 1 and 11 (d) 2 and 10
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
  25. 25.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 10201 (d) 9999
  26. 26.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 361 (d) 399
  27. 27.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  28. 28.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
  29. 29.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 22 (c) 13 (d) 16
  30. 30.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 10 (d) 9
  31. 31.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 361 (d) 185
  32. 32.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
  33. 33.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 7 (d) 8
  34. 34.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2209 (c) 2819 (d) 2809
  35. 35.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
  36. 36.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
  37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 333 (d) 657
  38. 38.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  39. 39.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 137 (c) 181 (d) 225
  40. 40.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 1 and 12 (d) 6 and 7

Answer key

  1. 4 and 8
  2. 17
  3. 4624
  4. 41
  5. (x + 7)(x + 8)
  6. 24
  7. 1521
  8. 6561
  9. 25
  10. (x + 4)(x + 5)
  11. (x + 5)(x + 7)
  12. 185
  13. (x + 4)(x + 8)
  14. 160
  15. 18
  16. 6241
  17. 4
  18. (x + 5)(x + 6)
  19. 181
  20. 5
  21. 25
  22. 61
  23. 5 and 7
  24. 10
  25. 9801
  26. 361
  27. (x + 4)(x + 6)
  28. (x + 2)(x + 9)
  29. 10
  30. 10
  31. 9
  32. 676
  33. 7
  34. 2809
  35. 4761
  36. 48
  37. 225
  38. 56
  39. 137
  40. 6 and 7

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