To complete the square for the polynomial \( g^2 + 12g \), we need to determine the number that, when added, will make it a perfect-square quadratic.
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Start with the given expression: \[ g^2 + 12g \]
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Take the coefficient of \( g \), which is 12, and halve it: \[ \frac{12}{2} = 6 \]
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Then, square this result: \[ 6^2 = 36 \]
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Now, add this squared value to the polynomial: \[ g^2 + 12g + 36 \]
Thus, the number that makes the polynomial a perfect-square quadratic is \( \boxed{36} \).
The complete expression can now be written as: \[ (g + 6)^2 \]
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