To solve the equation \(x^2 + 54x = -729\) using the perfect square trinomial pattern, we start by moving \(-729\) to the left side of the equation:
\[ x^2 + 54x + 729 = 0 \]
Next, we aim to rewrite the left side as a perfect square trinomial. The perfect square trinomial is expressed as \((x + a)^2\). To find \(a\), we take the coefficient of \(x\), which is \(54\), divide it by \(2\), and then square it:
\[ a = \frac{54}{2} = 27 \]
Then we compute \(a^2\):
\[ 27^2 = 729 \]
Thus, we can write the left-hand side of the equation as:
\[ (x + 27)^2 = 729 \]
Next, we take the square root of both sides:
\[ x + 27 = \pm \sqrt{729} \]
Calculating the square root gives:
\[ \sqrt{729} = 27 \]
So we have two cases to consider:
- \(x + 27 = 27\)
- \(x + 27 = -27\)
Now we solve each case:
- For \(x + 27 = 27\):
\[ x = 27 - 27 = 0 \]
- For \(x + 27 = -27\):
\[ x = -27 - 27 = -54 \]
Thus, the solutions to the equation are:
\[ x = 0 \quad \text{and} \quad x = -54 \]
So, the final answer is:
\[ x = 0 \quad \text{or} \quad x = -54 \]
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