To complete the square for the polynomial \(d^2 + 6d\), we need to find a constant that will make it a perfect-square quadratic.
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Take the coefficient of \(d\) (which is 6), divide it by 2, and then square the result: \[ \left(\frac{6}{2}\right)^2 = 3^2 = 9. \]
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Add this square to the polynomial: \[ d^2 + 6d + 9. \]
Now, the polynomial can be written as a perfect square: \[ (d + 3)^2. \]
Therefore, the number you fill in to make the polynomial \(d^2 + 6d\) a perfect-square quadratic is 9.
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