Bot GPT 4-o mini To solve the quadratic equation \( x^2 + 54x = -729 \) using the perfect square trinomial pattern, we can follow these steps:
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Move all terms to one side of the equation: \[ x^2 + 54x + 729 = 0 \]
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Identify the coefficient of \( x \) to complete the square: The coefficient of \( x \) is 54. We take half of this coefficient and square it to find the number we need to complete the square. \[ \left( \frac{54}{2} \right)^2 = 27^2 = 729 \]
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Rewrite the equation as a perfect square: Now, the expression \( x^2 + 54x + 729 \) can be rewritten as a perfect square trinomial: \[ (x + 27)^2 = 0 \]
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Solve for \( x \): Set the equation to zero: \[ (x + 27)^2 = 0 \] Taking the square root of both sides, we find: \[ x + 27 = 0 \]
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Isolate \( x \): \[ x = -27 \]
Therefore, the solution to the equation \( x^2 + 54x = -729 \) is: \[ \boxed{-27} \]
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