For what value of x, if any , is the equation `(x -1)^(2) =(x-7)^(2)` true ?

Options

A. `-4`
B. `-1`
C. 0
D. 4

Answer

To solve the equation \((x - 1)^{2} = (x - 7)^{2}\), we will follow these steps:

### Step 1: Expand both sides of the equation
We start by expanding both sides of the equation using the identity \((A - B)^{2} = A^{2} - 2AB + B^{2}\).

\[ (x - 1)^{2} = x^{2} - 2 \cdot 1 \cdot x + 1^{2} = x^{2} - 2x + 1 \]
\[ (x - 7)^{2} = x^{2} - 2 \cdot 7 \cdot x + 7^{2} = x^{2} - 14x + 49 \]

### Step 2: Set the expanded forms equal to each other
Now we have:
\[ x^{2} - 2x + 1 = x^{2} - 14x + 49 \]

### Step 3: Simplify the equation
Next, we will subtract \(x^{2}\) from both sides:
\[ -2x + 1 = -14x + 49 \]

### Step 4: Rearrange the equation
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ -2x + 14x = 49 - 1 \]
\[ 12x = 48 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 12 to isolate \(x\):
\[ x = \frac{48}{12} = 4 \]

### Conclusion
Thus, the value of \(x\) that satisfies the equation \((x - 1)^{2} = (x - 7)^{2}\) is:
\[ \boxed{4} \]

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