SOLUTION: Find all complex solutions to the equation z^4 = -64 - 16z^2. Enter all the solutions in the form $a + bi$, where $a$ and $b$ are real numbers.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find all complex solutions to the equation z^4 = -64 - 16z^2. Enter all the solutions in the form $a + bi$, where $a$ and $b$ are real numbers.      Log On


   

Question 1209643: Find all complex solutions to the equation z^4 = -64 - 16z^2.

Enter all the solutions in the form $a + bi$, where $a$ and $b$ are real numbers.
Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
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Find all complex solutions to the equation z^4 = -64 - 16z^2.
Enter all the solutions in the form a + bi, where a and b are real numbers.
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An equation

    z^4 = -64 - 16z^2

is equivalent to

    z^4 + 16z^2 + 64 = 0,

    %28z%5E2+%2B+8%29%5E2 = 0.


Over complex numbers, the last equation can be factored

    %28%28z+%2B+2i%2Asqrt%282%29%29%2A%28z+-+2i%2Asqrt%282%29%29%29%5E2 = 0,

or

    %28z%2B2i%2Asqrt%282%29%29%5E2%2A%28z-2i%2Asqrt%282%29%29%5E2 = 0.


It means that the given equation has these 4 roots

    z = -2i%2Asqrt%282%29  of the multiplicity 2,  and  z = 2i%2Asqrt%282%29  of the multiplicity 2.    ANSWER

Solved.