To complete the square for the polynomial \( p^2 + 13p \), we need to find the number that makes this expression a perfect-square quadratic.
- Take the coefficient of \( p \), which is \( 13 \).
- Divide it by \( 2 \): \[ \frac{13}{2} = 6.5 \]
- Square this result: \[ (6.5)^2 = 42.25 \]
Thus, the number we need to add to \( p^2 + 13p \) to complete the square is \( 42.25 \).
The completed expression will be: \[ p^2 + 13p + 42.25 \]
This can be rewritten as: \[ (p + 6.5)^2 \]
So, the final form is: \[ p^2 + 13p + 42.25 = (p + 6.5)^2 \]
To fill in the blank: \( p^2 + 13p + \text{42.25} \).
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