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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 415 · 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.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
  2. 2.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 1 and 8 (d) 4 and 5
  3. 3.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 25 (d) 73
  4. 4.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)
  5. 5.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
  6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 56 (d) 57
  7. 7.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 74 (d) 109
  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.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
  10. 10.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
  11. 11.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
  12. 12.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)
  13. 13.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 262 (d) 163
  14. 14.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  15. 15.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 129 (c) 89 (d) 249
  16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
  18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
  19. 19.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 9 (c) 3 (d) 169
  20. 20.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 8 (c) 100 (d) 82
  21. 21.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 26 (c) 4 (d) 36
  22. 22.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)
  23. 23.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  25. 25.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 49 (d) 169
  26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
  27. 27.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
  28. 28.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 8 (d) 5
  29. 29.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
  30. 30.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x − 3)(x − 9)
  31. 31.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 45 (c) 14 (d) 44
  32. 32.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 2 and 3 (c) 1 and 6 (d) 1 and 4
  33. 33.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 100 (c) 6 (d) 36
  34. 34.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)
  35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
  36. 36.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 9614 (d) 10404
  37. 37.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 9 (d) 11
  38. 38.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6399 (d) 6561
  39. 39.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8464 (c) 8096 (d) 7744
  40. 40.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 22 (c) 16 (d) 13

Answer key

  1. 729
  2. 4 and 5
  3. 25
  4. (x + 4)(x + 5)
  5. 63
  6. 56
  7. 74
  8. (x + 7)(x + 9)
  9. 48
  10. (x + 4)(x + 9)
  11. 6 and 7
  12. (x + 6)(x + 8)
  13. 130
  14. 2 and 8
  15. 89
  16. (x + 6)(x + 9)
  17. 441
  18. 17
  19. 9
  20. 64
  21. 16
  22. (x + 6)(x + 7)
  23. (x + 5)(x + 8)
  24. (x + 3)(x + 6)
  25. 49
  26. (x + 3)(x + 7)
  27. 15
  28. 6
  29. 17
  30. (x + 3)(x + 9)
  31. 45
  32. 2 and 3
  33. 36
  34. (x + 2)(x + 9)
  35. 208
  36. 9604
  37. 10
  38. 6241
  39. 7744
  40. 10

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