To complete the square for the polynomial \( u^2 - 14u \), we need to find a number that makes it a perfect-square quadratic.
The general form of completing the square is: \[ u^2 - 14u + \left(\frac{b}{2}\right)^2 \]
where \( b \) is the coefficient of \( u \) (which is \(-14\) in this case).
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First, calculate \(\frac{b}{2}\): \[ \frac{-14}{2} = -7 \]
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Now, square this value: \[ (-7)^2 = 49 \]
Thus, the number that makes the polynomial \( u^2 - 14u \) a perfect-square quadratic is \( 49 \).
So, the completed square form of the polynomial is: \[ u^2 - 14u + 49 = (u - 7)^2 \]
To summarize, you fill in the blank with: \[ \boxed{49} \]
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