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

Question

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

A. -3
B. -2
C. 2
D. 3
E. 6

WNBIB · Accepted Answer

Here is a logical approach to this question.
You see that the square of some number is equal to the absolute value of that number. This can only be attainable when that number is either 0 or 1. 
x - 2 = 0 or x-2 = 1
x = 2 or x = 3
Product of roots is 6

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VivekSri · Answer

OA is E

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btsaami · Answer

What's the Official solution for this?

piermacabj · Answer

If the square of a number is the number itself then the only possible solutions are 0 or 1

We know, by av properties, that |a|^2 = a^2

Therefore;

|x - 2|^2 = (x - 2)^2

So we end with the following equation:

|x - 2|^2 = |x - 2|

The number |x - 2| has to be either 1 or 0 so:

|x - 2| = 0

x = 2

|x - 2| = 1

x = 1 
x = 3

The roots are 1, 2 and 3, which product is 6
 Ans E

luisdicampo · Answer

Deconstructing the Question

We are asked for the product of all roots of  (x - 2)^2 = |x - 2| .

Let   y = x - 2  .

Then the equation becomes  y^2 = |y| .

Step-by-step

If  y \ge 0 , then  |y| = y .

So  y^2 = y .

Thus,

y(y - 1) = 0

So  y = 0  or  y = 1 .

If  y < 0 , then  |y| = -y .

So  y^2 = -y .

Thus,

y(y + 1) = 0

So  y = 0  or  y = -1 .

Only   -1   works in this case.

Therefore, the solutions for  y  are

y = -1, 0, 1

Now convert back to  x :

x = y + 2

So,

x = 1, 2, 3

Their product is

1 \cdot 2 \cdot 3 = 6

Answer: E

egmat · Answer

Here's a clean step-by-step solution:

The equation is (x -  2 )2 = |x -  2 |.

Step 1: Simplify with a substitution. 
Let u = x -  2 . The equation becomes:
u2 = |u|

Step 2: Rearrange. 
u2 - |u| =  0 
|u|(|u| -  1 ) =  0

(This works because u2 = |u|2, so we can factor out |u|.)

Step 3: Solve. 
Either |u| =  0 , which gives u =  0 
Or |u| -  1  =  0 , which gives |u| =  1 , so u =  1  or u =  -1

Step 4: Convert back to x  (since u = x -  2 ):
- u =  0  → x =  2 
- u =  1  → x =  3 
- u =  -1  → x =  1

Step 5: Find the product of all roots. 
 2  ×  3  ×  1  =  6

Answer: E

Key Insight:   When you see the same expression on both sides of an equation (here, x -  2 ),  substitute a single variable for it . This dramatically simplifies the algebra. Also,  never divide both sides by a variable (like |u|) — you'd lose the u = 0 solution .  Always factor instead.

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

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

Prep Toolkit

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