To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first rearrange the equation to set it to zero:
\[ x^2 + 54x + 729 = 0 \]
Next, we want to use the method of completing the square. To complete the square for the expression \( x^2 + 54x \), we follow these steps:
- Take the coefficient of \( x \), which is 54, divide it by 2, and then square it:
\[ \left(\frac{54}{2}\right)^2 = 27^2 = 729 \]
- We can express \( x^2 + 54x + 729 \) as a perfect square:
\[ (x + 27)^2 = 729 \]
This step gives us:
\[ (x + 27)^2 - 729 = 0 \]
Alternatively, we can write it as:
\[ (x + 27)^2 = 729 \]
- Next, we take the square root of both sides:
\[ x + 27 = \pm \sqrt{729} \]
Since \( \sqrt{729} = 27 \), we have:
\[ x + 27 = 27 \quad \text{or} \quad x + 27 = -27 \]
- Solving each equation gives us:
For \( x + 27 = 27 \):
\[ x = 27 - 27 = 0 \]
For \( x + 27 = -27 \):
\[ x = -27 - 27 = -54 \]
Thus, the solutions to the equation \( x^2 + 54x + 729 = 0 \) are:
\[ \boxed{0 \text{ and } -54} \]