Solve (x-1)(x-7)=-9.

Answer

To solve the equation \((x-1)(x-7)=-9\), we will follow these steps:

### Step 1: Expand the left-hand side
First, we expand the left-hand side of the equation:
\[ (x-1)(x-7) = x^2 - 7x - x + 7 = x^2 - 8x + 7 \]
So, we rewrite the equation as:
\[ x^2 - 8x + 7 = -9 \]

### Step 2: Move all terms to one side
Next, we move \(-9\) to the left-hand side by adding \(9\) to both sides:
\[ x^2 - 8x + 7 + 9 = 0 \]
This simplifies to:
\[ x^2 - 8x + 16 = 0 \]

### Step 3: Factor the quadratic equation
Now, we can factor the quadratic equation:
\[ x^2 - 8x + 16 = (x - 4)(x - 4) = (x - 4)^2 \]

### Step 4: Set the factored equation to zero
Next, we set the factored form equal to zero:
\[ (x - 4)^2 = 0 \]

### Step 5: Solve for \(x\)
Taking the square root of both sides gives us:
\[ x - 4 = 0 \]
Thus, solving for \(x\) yields:
\[ x = 4 \]

### Final Answer
The solution to the equation \((x-1)(x-7)=-9\) is:
\[ \boxed{4} \]