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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 343 · 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 48²? (a) 2704 (b) 2496 (c) 2314 (d) 2304
  2. 2.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6399 (c) 6241 (d) 6251
  3. 3.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 3 and 5 (d) 1 and 7
  4. 4.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 1 (d) 61
  5. 5.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 10 (d) 9
  6. 6.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
  7. 7.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 11 (c) 9 (d) 10
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  9. 9.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 17 (d) 11
  10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  11. 11.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 16 (d) 56
  12. 12.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10609 (c) 10619 (d) 9409
  13. 13.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
  14. 14.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 1 (d) 85
  15. 15.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1444 (d) 1764
  16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3625 (d) 3025
  18. 18.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  19. 19.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 104 (d) 144
  20. 20.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 3 (c) 49 (d) 29
  21. 21.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  22. 22.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 44 (d) 46
  23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 361 (d) 25
  24. 24.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  25. 25.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 12 (c) 11 (d) 8
  26. 26.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
  27. 27.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 91 (c) 181 (d) 121
  28. 28.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3035 (d) 3025
  29. 29.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 2 and 5 (c) 1 and 6 (d) 3 and 4
  30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3969 (d) 3979
  32. 32.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 5 and 9 (d) 1 and 13
  33. 33.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
  34. 34.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 9 (c) 3 (d) 81
  35. 35.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 1 and 8 (c) 4 and 5 (d) 2 and 10
  36. 36.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 185 (c) 361 (d) 273
  39. 39.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
  40. 40.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1764 (d) 1454

Answer key

  1. 2304
  2. 6241
  3. 3 and 5
  4. 1
  5. 10
  6. (x + 4)(x + 8)
  7. 10
  8. (x + 6)(x + 8)
  9. 18
  10. (x + 7)(x + 8)
  11. 15
  12. 10609
  13. 1
  14. 1
  15. 1764
  16. (x + 6)(x + 9)
  17. 4225
  18. 16
  19. 104
  20. 9
  21. (x + 6)(x + 7)
  22. 45
  23. 25
  24. 289
  25. 12
  26. 202
  27. 61
  28. 3025
  29. 2 and 5
  30. (x + 5)(x + 9)
  31. 3969
  32. 5 and 9
  33. 784
  34. 9
  35. 4 and 5
  36. (x + 2)(x + 3)
  37. (x + 7)(x + 9)
  38. 185
  39. 208
  40. 1444

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