To solve the equation \( x^2 - 16x - 13 = 4 \) by completing the square, we start by rearranging the equation to set it to zero:
\[ x^2 - 16x - 13 - 4 = 0 \]
This simplifies to:
\[ x^2 - 16x - 17 = 0 \]
Next, we can move the constant term to the other side:
\[ x^2 - 16x = 17 \]
Now, we complete the square. To do this, take half of the coefficient of \( x \) (which is \(-16\)), square it, and add it to both sides. Half of \(-16\) is \(-8\), and squaring it gives \(64\):
\[ x^2 - 16x + 64 = 17 + 64 \]
This simplifies to:
\[ (x - 8)^2 = 81 \]
Now, we take the square root of both sides:
\[ x - 8 = \pm 9 \]
This results in two equations:
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\( x - 8 = 9 \) \ \( x = 9 + 8 = 17 \)
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\( x - 8 = -9 \) \ \( x = -9 + 8 = -1 \)
Thus, the two values that solve the equation are:
\[ x = 17 \quad \text{and} \quad x = -1 \]
So, the final answers for \( x \) are:
\[ x = 17, \quad x = -1 \]