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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 520 · 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 + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  2. 2.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 28 (c) 11 (d) 10
  3. 3.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 2 and 6 (c) 1 and 7 (d) 1 and 15
  4. 4.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 37 (c) 61 (d) 49
  5. 5.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 6 (c) 68 (d) 36
  6. 6.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
  7. 7.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 13 (c) 45 (d) 15
  8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 7225 (d) 8125
  12. 12.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 14 (d) 48
  13. 13.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 7 (d) 10
  14. 14.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 49 (c) 39 (d) 29
  15. 15.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
  16. 16.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
  17. 17.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  20. 20.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 52 (d) 100
  21. 21.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 15 (c) 14 (d) 9
  22. 22.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 196 (d) 106
  23. 23.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 4761 (d) 5051
  24. 24.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)
  25. 25.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 4 (d) 400
  26. 26.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  27. 27.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
  28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 14 (c) 16 (d) 54
  29. 29.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 7 (d) 8
  30. 30.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 289 (c) 223 (d) 421
  31. 31.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  32. 32.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
  33. 33.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 352 (d) 160
  34. 34.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  35. 35.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)
  36. 36.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 391 (d) 299
  38. 38.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
  39. 39.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 49 (c) 7 (d) 85
  40. 40.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 74 (d) 109

Answer key

  1. (x + 5)(x + 7)
  2. 11
  3. 3 and 5
  4. 37
  5. 36
  6. 181
  7. 14
  8. (x + 7)(x + 9)
  9. (x + 4)(x + 5)
  10. (x + 3)(x + 6)
  11. 9025
  12. 14
  13. 7
  14. 29
  15. 1444
  16. 4
  17. (x + 5)(x + 9)
  18. (x + 6)(x + 7)
  19. (x + 8)(x + 9)
  20. 52
  21. 14
  22. 106
  23. 5041
  24. (x + 3)(x + 7)
  25. 16
  26. 146
  27. 1681
  28. 15
  29. 7
  30. 157
  31. 72
  32. 48
  33. 160
  34. 15
  35. (x + 7)(x + 8)
  36. (x + 6)(x + 9)
  37. 289
  38. 441
  39. 49
  40. 74

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