To solve the quadratic equation \(x^2 - 16x - 13 = 4\) by completing the square, we first rearrange the equation:
\[ x^2 - 16x - 13 - 4 = 0 \]
This simplifies to:
\[ x^2 - 16x - 17 = 0 \]
Next, we will complete the square for the expression \(x^2 - 16x\). To do this, we take the coefficient of \(x\), which is \(-16\), divide it by 2 to get \(-8\), and then square it to get \(64\).
Now, we add and subtract \(64\) inside the equation:
\[ x^2 - 16x + 64 - 64 - 17 = 0 \]
This simplifies to:
\[ (x - 8)^2 - 81 = 0 \]
Next, we isolate the square:
\[ (x - 8)^2 = 81 \]
Now, we'll take the square root of both sides:
\[ x - 8 = 9 \quad \text{or} \quad x - 8 = -9 \]
Solving for \(x\) in each case gives:
- \(x - 8 = 9\) leads to \(x = 17\).
- \(x - 8 = -9\) leads to \(x = -1\).
Thus, the two values that solve the quadratic equation \(x^2 - 16x - 13 = 4\) are:
\[ \boxed{17 \text{ and } -1} \]
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