To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we first rearrange the equation to bring all terms to one side:
\[ x^2 + 54x + 729 = 0 \]
Next, we want to express the left-hand side as a perfect square trinomial. To do this, we can complete the square.
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Start with the coefficient of \( x \), which is \( 54 \). Take half of \( 54 \) and square it: \[ \left(\frac{54}{2}\right)^2 = 27^2 = 729 \]
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The expression \( x^2 + 54x + 729 \) can then be rewritten as: \[ (x + 27)^2 = 729 \]
Next, we solve for \( x + 27 \):
\[ (x + 27)^2 = 729 \]
Taking the square root of both sides gives:
\[ x + 27 = \pm 27 \]
Now we can solve for \( x \):
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For the positive case: \[ x + 27 = 27 \] \[ x = 27 - 27 = 0 \]
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For the negative case: \[ x + 27 = -27 \] \[ x = -27 - 27 = -54 \]
Thus, the solutions to the equation \( x^2 + 54x = -729 \) are:
\[ x = 0 \quad \text{and} \quad x = -54 \]
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