Let's fill in the steps to complete the square for the equation \( x^2 + 12x - 14 = 0 \).
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Start with the original equation: \[ x^2 + 12x - 14 = 0 \]
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Move the constant term to the right side: \[ x^2 + 12x = 14 \]
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To complete the square, take half of the coefficient of \( x \) (which is 12), square it, and add it to both sides. Half of 12 is 6, and \( 6^2 = 36 \): \[ x^2 + 12x + 36 = 14 + 36 \]
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This simplifies to: \[ x^2 + 12x + 36 = 50 \]
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We can now factor the left-hand side: \[ (x + 6)^2 = 50 \]
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At this point, we need to take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \] (Note: \( \sqrt{50} = 5\sqrt{2} \))
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After that is done, we will have: \[ x + 6 = 5\sqrt{2} \quad \text{or} \quad x + 6 = -5\sqrt{2} \]
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To isolate the variable \( x \), we subtract 6 from both sides: \[ x = -6 + 5\sqrt{2} \quad \text{or} \quad x = -6 - 5\sqrt{2} \]
Final answers are: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]