MathRevolution wrote:

[GMAT math practice question]

What is the product of all the roots on the equation (x-2)^2=|x-2|?

A. 2

B. -2

C. 3

D. -3

E. 6

Case 1: If \(x-2>0\), then \((x-2)^2=(x-2)\)

or \((x-2)^2-(x-2)=0 =>(x-2)(x-2-1)=0\)

Hence the roots are \(x-2=0\), or \(x=2\) and \(x-3=0\), or \(x=3\)

Case 2 If \(x-2<0\), then \((x-2)^2=-(x-2)\)

or \((x-2)^2+(x-2)=0 =>(x-2)(x-2+1)=0\)

Hence the roots are \(x-2=0\), or \(x=2\) and \(x-1=0\), or \(x=1\)

So, combining Case 1 & Case 2, the roots of the equation are \(1\), \(2\) and \(3\)

Product of roots \(=1*2*3=6\)

Option

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Another method

\((x-2)^2=|x-2|\), square both sides to get \((x-2)^4=(x-2)^2\)

or \((x-2)^4-(x-2)^2=0 =>(x-2)^2(x-3)(x-1)=0\)

So roots of the equation are \((x-2)^2=0\), \(x-3=0\) and \(x-1=0\)

Hence \(x = 1\), \(2\) and \(3\)

So Product of roots \(= 1*2*3\)