To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first move all terms to one side of the equation:
\[ x^2 + 54x + 729 = 0 \]
Next, we will complete the square. The first step in this process is to take half of the coefficient of \( x \) (which is 54), square it, and add it to both sides.
- Half of 54 is \( \frac{54}{2} = 27 \).
- Squaring 27 gives us \( 27^2 = 729 \).
Then we rewrite the equation incorporating this square:
\[ x^2 + 54x + 729 = 729 \]
This can be rewritten as:
\[ (x + 27)^2 = 729 \]
Now we can take the square root of both sides:
\[ x + 27 = \pm \sqrt{729} \]
Calculating the square root of 729 gives us:
\[ \sqrt{729} = 27 \]
So we have:
\[ x + 27 = 27 \quad \text{or} \quad x + 27 = -27 \]
Now we 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 to the quadratic equation are:
\[ \boxed{0} \quad \text{and} \quad \boxed{-54} \]
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