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To solve the quadratic equation \((x - 4)^2 = 25\) using square roots, follow these steps:
1. Take the square root of both sides of the equation:
\[
x - 4 = \pm 5
\]
2. Solve for \(x\) by separating the two cases:
**Case 1:**
\[
x - 4 = 5 \\
x = 5 + 4 = 9
\]
**Case 2:**
\[
x - 4 = -5 \\
x = -5 + 4 = -1
\]
Thus, the solutions are:
\[
x = 9 \quad \text{and} \quad x = -1
\]
So the answer is \(\{-1, 9\}\).