If `|2x - 4|` is equal to 2 and `(x - 3)^2` is equal to 4, then what is the value of x?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the given problem, we need to address two equations:

1. \( |2x - 4| = 2 \)
2. \( (x - 3)^2 = 4 \)

Let's solve each equation step by step.

### Step 1: Solve the first equation \( |2x - 4| = 2 \)

The absolute value equation \( |A| = B \) implies two cases:

**Case 1:** \( 2x - 4 = 2 \)

**Case 2:** \( 2x - 4 = -2 \)

#### Case 1:
1. \( 2x - 4 = 2 \)
2. Add 4 to both sides:
\[ 2x = 2 + 4 \]
\[ 2x = 6 \]
3. Divide by 2:
\[ x = 3 \]

#### Case 2:
1. \( 2x - 4 = -2 \)
2. Add 4 to both sides:
\[ 2x = -2 + 4 \]
\[ 2x = 2 \]
3. Divide by 2:
\[ x = 1 \]

From the first equation, we find two possible values for \( x \): \( x = 3 \) and \( x = 1 \).

### Step 2: Solve the second equation \( (x - 3)^2 = 4 \)

To solve this, we take the square root of both sides:

1. \( x - 3 = 2 \) or \( x - 3 = -2 \)

#### Sub-case 1:
1. \( x - 3 = 2 \)
2. Add 3 to both sides:
\[ x = 2 + 3 \]
\[ x = 5 \]

#### Sub-case 2:
1. \( x - 3 = -2 \)
2. Add 3 to both sides:
\[ x = -2 + 3 \]
\[ x = 1 \]

From the second equation, we find two possible values for \( x \): \( x = 5 \) and \( x = 1 \).

### Step 3: Combine results from both equations

From the first equation, we found \( x = 3 \) and \( x = 1 \).
From the second equation, we found \( x = 5 \) and \( x = 1 \).

The common value from both sets of solutions is \( x = 1 \).

### Final Answer:
The value of \( x \) is \( \boxed{1} \).

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