To complete the square for the polynomial \( z^2 - 12z \), we need to take the coefficient of the \( z \) term, which is \(-12\), divide it by \( 2 \), and square the result.
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Take the coefficient of \( z \): \[ -12 \]
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Divide by \( 2 \): \[ -12 / 2 = -6 \]
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Square the result: \[ (-6)^2 = 36 \]
Now, we add \( 36 \) to the polynomial. Thus, the completed square form of the polynomial is:
\[ z^2 - 12z + 36 = (z - 6)^2 \]
The number that makes the polynomial a perfect-square quadratic is \( 36 \).