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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 731 · 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 + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
  2. 2.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 4 (d) 170
  3. 3.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
  4. 4.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
  5. 5.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 4 (d) 130
  6. 6.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)
  7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  8. 8.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
  9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
  11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  12. 12.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 14 (c) 45 (d) 46
  13. 13.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  14. 14.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 324 (d) 484
  15. 15.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 40 (d) 1 and 12
  16. 16.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 5051 (d) 4901
  17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 15 (c) 48 (d) 14
  18. 18.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 2 and 9 (c) 1 and 18 (d) 3 and 6
  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 = 11 and ab = 18, what is a² + b²? (a) 103 (b) 121 (c) 157 (d) 85
  21. 21.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
  22. 22.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6094 (d) 6396
  23. 23.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 31 (c) 12 (d) 33
  24. 24.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2809 (c) 2509 (d) 2819
  25. 25.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  26. 26.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)
  27. 27.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 4 and 5 (d) 1 and 20
  28. 28.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 7 (d) 10
  29. 29.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  30. 30.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 9 (d) 11
  31. 31.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 81 (d) 9
  32. 32.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1849 (d) 1379
  33. 33.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
  34. 34.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 441 (d) 9
  35. 35.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 202 (d) 400
  36. 36.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)
  37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 784 (d) 1034
  39. 39.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 1 and 18 (c) 2 and 9 (d) 3 and 6
  40. 40.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185

Answer key

  1. (x + 4)(x + 8)
  2. 16
  3. (x + 2)(x + 3)
  4. 1
  5. 4
  6. (x + 5)(x + 8)
  7. (x + 6)(x + 7)
  8. 1444
  9. (x + 5)(x + 9)
  10. 6889
  11. (x + 6)(x + 8)
  12. 45
  13. (x + 4)(x + 7)
  14. 164
  15. 5 and 8
  16. 5041
  17. 14
  18. 2 and 9
  19. (x + 8)(x + 9)
  20. 85
  21. 13
  22. 6084
  23. 32
  24. 2809
  25. 16
  26. (x + 7)(x + 8)
  27. 4 and 5
  28. 10
  29. 63
  30. 10
  31. 81
  32. 1369
  33. (x + 2)(x + 7)
  34. 9
  35. 202
  36. (x + 7)(x + 9)
  37. (x + 5)(x + 7)
  38. 1024
  39. 3 and 6
  40. 185

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