What is the process of solving an equation with the square and the 4th power of an unknown, for example x^4−6x^2−27=0?

Haiphongum 2022-09-23 Answered
What is the process of solving an equation with the square and the 4th power of an unknown, for example x 4 6 x 2 27 = 0
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Answers (1)

Lilliana Mason
Answered 2022-09-24 Author has 11 answers
This is a quadratic in terms of x 2 . I know solutions have already been posted, but this is the way I like to solve these problems:
x 4 6 x 2 27 = ( x 2 9 ) ( x 2 + 3 )
If we are solving: ( x 2 9 ) ( x 2 + 3 ) = 0, this gives us two equations:
x 2 9 = 0 or x 2 + 3 = 0
The one on the left yields:
x 2 = 9 x = ± 3
The one on the right yields:(where i is the imaginary unit)
x 2 = 3 x = ± 3 i
These are the four roots promised us by the Fundamental Theorem of Algebra. And, as expected, the complex roots came in conjugate pairs.
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