Problem Solving

[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

Max@Math Revolution wrote:[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.

Max@Math Revolution wrote:[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

If we replace x-1 with a, we get:
a² = |a|.
The resulting equation is valid only if a -- in other words, the value of x-1 -- is equal to -1, 0, or 1:
x-1 = -1 --> x=0
x-1 = 0 ---> x=1
x-1 = 1 ---> x=2
Sum of the solutions = 0+1+2 = 3.

The correct answer is E.

=>

(x-1)^2=|x-1|
=> |x-1|^2=|x-1|
=> |x-1|^2-|x-1|=0
=> |x-1| (|x-1|-1)=0
=> |x-1| = 0 or |x-1|-1=0
=> |x-1| = 0 or |x-1|=1
=> x-1 = 0 or x-1=±1
=> x=1 = 0 or x=1±1
=> x=1, x=0 or x=2

The sum of the solutions is 0 + 1 + 2 = 3.

Therefore, the answer is E.
Answer: E