To complete the square for the quadratic expression \( w^2 - 21w \), we need to take the coefficient of \( w \), which is \(-21\), divide it by \( 2 \), and then square the result.
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Divide the coefficient by 2: \[ \frac{-21}{2} = -\frac{21}{2} \]
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Square the result: \[ \left(-\frac{21}{2}\right)^2 = \frac{441}{4} \]
Thus, by adding \( \frac{441}{4} \) to the expression, we can write:
\[ w^2 - 21w + \frac{441}{4} \]
This expression is a perfect square and can be factored as:
\[ \left(w - \frac{21}{2}\right)^2 \]
Therefore, the number that makes the polynomial \( w^2 - 21w \) a perfect-square quadratic is \( \frac{441}{4} \).
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