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 of the equation: \[ x^2 + 12x = 14 \]
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Take the coefficient of \( x \), which is 12, divide it by 2, and then square it: \[ \left(\frac{12}{2}\right)^2 = 6^2 = 36 \]
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Add this square to both sides of the equation: \[ x^2 + 12x + 36 = 14 + 36 \] Simplifying the right side: \[ x^2 + 12x + 36 = 50 \]
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Rewrite the left side as a squared binomial: \[ (x + 6)^2 = 50 \]
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Take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \]
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Solve for \( x \) by isolating it: \[ x = -6 \pm \sqrt{50} \]
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Simplify \(\sqrt{50}\): \[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \]
So the solutions can be expressed as: \[ x = -6 \pm 5\sqrt{2} \]
Thus, the solutions to the equation \( x^2 + 12x - 14 = 0 \) are: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]
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