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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 540 · 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 + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  2. 2.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
  3. 3.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2916 (d) 2926
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
  5. 5.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
  6. 6.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 13 (d) 22
  7. 7.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 916 (d) 1156
  10. 10.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
  11. 11.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 5 and 7 (d) 1 and 35
  12. 12.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5476 (d) 5486
  13. 13.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
  14. 14.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  16. 16.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 48 (d) 14
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
  18. 18.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  19. 19.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
  20. 20.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 12 (c) 27 (d) 28
  21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
  22. 22.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
  23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12
  24. 24.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 40 (c) 41 (d) 13
  25. 25.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1859 (d) 1849
  26. 26.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 896 (c) 1024 (d) 784
  28. 28.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 208 (d) 304
  29. 29.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 1 and 24 (d) 3 and 8
  30. 30.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  31. 31.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  32. 32.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 52 (d) 148
  33. 33.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
  34. 34.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 35 (d) 34
  35. 35.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
  36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  37. 37.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 252 (d) 180
  38. 38.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 109 (d) 169
  39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 441 (d) 361
  40. 40.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 4 (d) 256

Answer key

  1. (x + 5)(x + 8)
  2. 14
  3. 2916
  4. 4
  5. 225
  6. 10
  7. (x + 5)(x + 6)
  8. (x + 6)(x + 9)
  9. 1156
  10. 2809
  11. 5 and 7
  12. 5476
  13. 8
  14. (x + 5)(x + 7)
  15. (x + 7)(x + 9)
  16. 14
  17. 17
  18. 181
  19. (x + 2)(x + 8)
  20. 27
  21. 72
  22. 6 and 7
  23. 13
  24. 40
  25. 1849
  26. (x + 6)(x + 8)
  27. 784
  28. 208
  29. 3 and 8
  30. 16
  31. 48
  32. 52
  33. 25
  34. 35
  35. 1296
  36. (x + 8)(x + 9)
  37. 180
  38. 109
  39. 441
  40. 16

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