MathRevolution wrote:

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Math Revolution GMAT math practice question]

What is the sum of the solutions of the equation \((x-1)^2=|x-1|?\)

\(A. -1\)

\(B. 0\)

\(C. 1\)

\(D. 2\)

\(E. 3\)

\(?\,\,\,:\,\,\,{\text{sum}}\,\,{\text{of}}\,\,{\text{roots}}\)

\({\left( {x - 1} \right)^2} = \left| {x - 1} \right|\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\left( {x - 1} \right)^4} = {\left( {x - 1} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}

\,\,\,x = 1\,\,\,\left( {{\text{trivial}}\,\,{\text{inspection}}} \right) \hfill \\

\,\,{\text{for}}\,\,x \ne 1\,\,:\,\,\,\,\,\frac{{{{\left( {x - 1} \right)}^4}}}{{{{\left( {x - 1} \right)}^2}}} = \frac{{{{\left( {x - 1} \right)}^2}}}{{{{\left( {x - 1} \right)}^2}}}\,\,\,\,\, \Rightarrow \,\,\,\,{\left( {x - 1} \right)^2} = 1\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 0\,\,\,{\text{or}}\,\,\,x = 2 \hfill \\

\end{gathered} \right.\)

(*) When we "square" one equation (or, in general, put it to an EVEN positive power), we don´t loose original roots, but we eventually "add" new ones.

That´s why, in the end, we must check each POTENTIAL root in the original equation! All of them fit here. Therefore:

\(? = 1 + 0 + 2 = 3\)

This solution follows the notations and rationale taught in the GMATH method.

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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