College Algebra, Chapter 1, 1.5, Section 1.5, Problem 54

Find all real solutions of the equation $\displaystyle 4x^4 - 16x^2 + 4 = 0$


$
\begin{equation}
\begin{aligned}

4x^4 - 16x^2 + 4 =& 0
&& \text{Given}
\\
\\
x^4 - 4x^2 + 1 =&0
&& \text{Divide both sides by } 4
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w^2 - 4w + 1 =& 0
&& \text{Let } w = x^2
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w^2 - 4w =& -1
&& \text{Subtract } 1
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w^2 - 4w + 4 =& -1 + 4
&& \text{Complete the square: add } \left( \frac{-4}{2} \right)^2 = 4
\\
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(w - 2)^2 =& 3
&& \text{Perfect square}
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w - 2 =& \pm \sqrt{3}
&& \text{Take the square root}
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w =& 2 \pm \sqrt{3}
&& \text{Solve for } w
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x^2 =& 2 \pm \sqrt{3}
&& \text{Substitute } w = x^2
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x =& \pm (2 \pm \sqrt{3})^{\frac{1}{2}}
&& \text{Solve for } x
\\
\\
X =& \pm (2 \pm \sqrt{3})^{\frac{1}{2}}
&&


\end{aligned}
\end{equation}
$