To complete the square for the polynomial \( x^2 + 18x + \), we need to find a constant that makes this expression a perfect square.
- Start with the quadratic part: \( x^2 + 18x \).
- Take the coefficient of \( x \) (which is 18), divide it by 2, and then square it:
\[ \left(\frac{18}{2}\right)^2 = 9^2 = 81 \]
- Now, we add and subtract this constant (81) to the polynomial:
\[ x^2 + 18x + 81 - 81 = (x + 9)^2 - 81 \]
Thus, the number that makes the polynomial a perfect-square quadratic is 81. Therefore:
\[ x^2 + 18x + 81 = (x + 9)^2 \]
In conclusion, the completed polynomial is \( x^2 + 18x + 81 \).
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