To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we can first rewrite the equation in a standard form.
- Move the constant term to the left side:
\[ x^2 + 54x + 729 = 0 \]
- Now, we want to complete the square for the left side \( x^2 + 54x + 729 \).
To complete the square:
- Take half of the coefficient of \( x \), which is \( 54 \), divide it by 2 to get \( 27 \).
- Square \( 27 \) to obtain \( 729 \).
This confirms we can rewrite the expression as a perfect square trinomial:
\[ x^2 + 54x + 729 = (x + 27)^2 \]
We can redefine this equation as:
\[ (x + 27)^2 = 0 \]
- Set the square equal to zero and solve for \( x \):
\[ x + 27 = 0 \]
\[ x = -27 \]
Thus, the solution to the quadratic equation \( x^2 + 54x = -729 \) is:
\[ \boxed{-27} \]