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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 642 · 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 79²? (a) 6241 (b) 6561 (c) 6399 (d) 6251
  2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  3. 3.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
  4. 4.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
  8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
  10. 10.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
  11. 11.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
  12. 12.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3969 (d) 3249
  13. 13.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 146 (d) 201
  14. 14.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 8291 (d) 7921
  15. 15.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 2 and 6 (d) 1 and 15
  16. 16.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
  17. 17.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)
  18. 18.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  19. 19.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
  20. 20.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 14 (d) 12
  21. 21.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 25 (d) 33
  22. 22.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 37 (c) 49 (d) 25
  23. 23.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 7744 (d) 8096
  24. 24.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1444 (d) 1764
  25. 25.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
  26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
  27. 27.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  28. 28.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
  29. 29.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
  30. 30.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
  31. 31.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 529 (b) 289 (c) 409 (d) 539
  32. 32.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 441 (d) 401
  33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
  34. 34.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
  35. 35.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  36. 36.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 11035 (d) 10025
  37. 37.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
  38. 38.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
  39. 39.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
  40. 40.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634

Answer key

  1. 6241
  2. (x + 5)(x + 7)
  3. 74
  4. (x + 4)(x + 6)
  5. 7056
  6. (x + 7)(x + 8)
  7. 2809
  8. 15
  9. 17
  10. (x + 3)(x + 5)
  11. 14
  12. 3969
  13. 146
  14. 8281
  15. 3 and 5
  16. 45
  17. (x + 6)(x + 9)
  18. 63
  19. 82
  20. 13
  21. 17
  22. 25
  23. 7744
  24. 1764
  25. (x + 3)(x + 6)
  26. (x + 2)(x + 7)
  27. (x + 2)(x + 8)
  28. 125
  29. 34
  30. 12
  31. 529
  32. 441
  33. 181
  34. 5
  35. (x + 5)(x + 8)
  36. 11025
  37. (x + 6)(x + 7)
  38. 1
  39. 16
  40. 4624

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