The roots of the equation `(2-x)^(2)=16` are

Options

A. `+-4`
B. `-6,+2`
C. `6,-2`
D. `6,2`

Answer

To find the roots of the equation \((2-x)^2 = 16\), we can follow these steps:

### Step 1: Take the square root of both sides
To eliminate the square, we take the square root of both sides of the equation:
\[ \sqrt{(2-x)^2} = \sqrt{16} \]
This simplifies to:
\[ 2 - x = \pm 4 \]

### Step 2: Solve for \(x\) using both cases
Now we have two cases to consider because of the \(\pm\) sign.

**Case 1:**
\[ 2 - x = 4 \]
To solve for \(x\), we subtract 2 from both sides:
\[ -x = 4 - 2 \]
\[ -x = 2 \]
Now, multiply both sides by -1:
\[ x = -2 \]

**Case 2:**
\[ 2 - x = -4 \]
Again, we subtract 2 from both sides:
\[ -x = -4 - 2 \]
\[ -x = -6 \]
Now, multiply both sides by -1:
\[ x = 6 \]

### Step 3: Write the final roots
The roots of the equation \((2-x)^2 = 16\) are:
\[ x = -2 \quad \text{and} \quad x = 6 \]

### Summary of the solution:
The roots of the equation are \(x = -2\) and \(x = 6\).

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