To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first want to move all terms to one side of the equation:
\[ x^2 + 54x + 729 = 0. \]
Next, we can complete the square. The first step is to take the coefficient of \( x \), which is 54, divide it by 2, and then square the result:
\[ \left(\frac{54}{2}\right)^2 = 27^2 = 729. \]
Now, we can rewrite the quadratic equation as a perfect square trinomial:
\[ (x + 27)^2 = 729. \]
Next, we take the square root of both sides:
\[ x + 27 = \pm \sqrt{729}. \]
Since \( \sqrt{729} = 27 \), we have:
\[ x + 27 = 27 \quad \text{or} \quad x + 27 = -27. \]
Now, we can solve for \( x \):
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For \( x + 27 = 27 \): \[ x = 27 - 27 = 0. \]
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For \( x + 27 = -27 \): \[ x = -27 - 27 = -54. \]
Thus, the solutions are:
\[ x = 0 \quad \text{or} \quad x = -54. \]
So, the final answer is:
\[ x = 0, -54. \]
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