To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first need to rearrange the equation so that one side is zero:
\[ x^2 + 54x + 729 = 0 \]
Next, we'll complete the square. We take the coefficient of \( x \) (which is 54), divide it by 2, and square it:
\[ \left( \frac{54}{2} \right)^2 = 27^2 = 729 \]
Since we already have \( +729 \) on the left side, we recognize that the left side can be expressed as a perfect square:
\[ (x + 27)^2 = 0 \]
Taking the square root of both sides gives us:
\[ x + 27 = 0 \]
Now, solving for \( x \):
\[ x = -27 \]
Thus, the solution to the equation \( x^2 + 54x = -729 \) is:
\[ \boxed{-27} \]
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