College Algebra, Chapter 1, 1.5, Section 1.5, Problem 56

Find all real solutions of the equation $\sqrt[3]{4x^2 - 4x} = x$


$
\begin{equation}
\begin{aligned}

\sqrt[3]{4x^2 - 4x} =& x
&& \text{Given}
\\
\\
4x^2 - 4x =& x^3
&& \text{Raise both sides of the equation by } 3
\\
\\
x^3 - 4x^2 + 4x =& 0
&& \text{Subtract } 4x^2 \text{ and add } 4x
\\
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x(x^2 - 4x + 1) =&0
&& \text{Factor out } x
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x =& 0 \text{ and } x^2 - 4x + 1 = 0
&& \text{Zero Product Property}
\\
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x^2 - 4x =& -1
&& \text{Subtract } 1
\\
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(x^2 - 4x + 4) =& -1 + 4
&& \text{Complete the square: add } \left( \frac{-4}{2} \right)^2 = 4
\\
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(x - 2)^2 =& 3
&& \text{Perfect Square}
\\
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x - 2 =& \pm \sqrt{3}
&& \text{Take the square root}
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x =& 2 \pm \sqrt{3}
&& \text{Solve for } x
\\
\\
x =& 0
&& \text{The only solution that satisfy the equation}

\end{aligned}
\end{equation}
$