# 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?

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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Lilliana Mason
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:
${x}^{4}-6{x}^{2}-27=\left({x}^{2}-9\right)\left({x}^{2}+3\right)$
If we are solving: $\left({x}^{2}-9\right)\left({x}^{2}+3\right)=0$, this gives us two equations:
${x}^{2}-9=0\phantom{\rule{2em}{0ex}}\text{or}\phantom{\rule{2em}{0ex}}{x}^{2}+3=0$
The one on the left yields:
${x}^{2}=9\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}x=±3$
The one on the right yields:(where i is the imaginary unit)
${x}^{2}=-3\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}x=±\sqrt{3}i$
These are the four roots promised us by the Fundamental Theorem of Algebra. And, as expected, the complex roots came in conjugate pairs.