Alg2/geo

Expanding  (a + bi)⁴  =  a⁴ + 4a³bi + 6a²b²i² + 4ab³i³ + b⁴i⁴  =  a⁴ + 4a³bi - 6a²b² - 4ab³i + b⁴

Since all the i-terms must cancel for the answer to be a real number:   4a³bi - 4ab³i - 24i  must have value 0.

   --->   4abi(a² - b²) - 24i  =  0

   --->   ab(a² - b²) - 6  =  0                 

   --->   ab(a² - b²) = 6 

Therefore, using factors of 6:  either  ab = 1  and a² - b² = 6  which is impossible because a = 1 and b = 1 and then a² - b² isn't 6

                                                      or  ab = 2  and a² - b² = 3  which means a = 2 and b = 1  -- this works!

                                                      or  ab = 3  and a² - b² = 2  which means a = 3 and b = 2 which is impossible

                                                      or  ab = 6  and  a² - b² = 1  which means a = 6 and b = 1 which is impossible

 

Now that you know that a = 2 and b = 1, place these back into the original formula   (a + bi)⁴ - 24i  to find the value of c.