To complete the square for the equation \( x^2 + 12x - 14 = 0 \), follow these steps:
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Move the constant term to the other side: \[ x^2 + 12x = 14 \]
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Take half of the coefficient of \(x\), which is \(12\), divide it by \(2\) to get \(6\), and then square it: \[ \left(\frac{12}{2}\right)^2 = 6^2 = 36 \]
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Add and subtract this square to the left side: \[ x^2 + 12x + 36 - 36 = 14 \] This simplifies to: \[ (x + 6)^2 - 36 = 14 \]
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Now, add \(36\) to both sides to balance the equation: \[ (x + 6)^2 = 14 + 36 \] \[ (x + 6)^2 = 50 \]
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Take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \] Simplifying \(\sqrt{50}\): \[ \sqrt{50} = 5\sqrt{2} \] Thus: \[ x + 6 = \pm 5\sqrt{2} \]
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Isolate \(x\): \[ x = -6 \pm 5\sqrt{2} \]
So the solutions to the equation \(x^2 + 12x - 14 = 0\) after completing the square are: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]