To complete the square for the equation \( x^2 + 12x - 14 = 0 \), we can follow these steps:
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Start with the equation: \[ x^2 + 12x - 14 = 0 \]
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Move the constant term to the other side: \[ x^2 + 12x = 14 \]
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To complete the square, we need to add a specific value to both sides of the equation. The value to add is found by taking half of the coefficient of \( x \) (which is 12), squaring it, and adding it to both sides. Half of 12 is 6, and squaring it gives us \( 6^2 = 36 \): \[ x^2 + 12x + 36 = 14 + 36 \]
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Rewrite the left hand side as a perfect square and the right hand side: \[ (x + 6)^2 = 50 \]
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Now that we have the equation \( (x + 6)^2 = 50 \), we take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \]
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Next, isolate \( x \) by subtracting 6: \[ x = -6 \pm \sqrt{50} \] Simplifying \( \sqrt{50} \) gives us \( \sqrt{25 \cdot 2} = 5\sqrt{2} \): \[ x = -6 \pm 5\sqrt{2} \]
So the final answers are: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]
In summary:
- \( x^2 + 12x - 14 = 0 \)
- \( x^2 + 12x = 14 \)
- \( x^2 + 12x + 36 = 14 + 36 \)
- We factor and simplify to get \( (x + 6)^2 = 50 \)
- At this point, we need to take the square root.
- After this is done, we will have \( x = -6 \pm 5\sqrt{2} \).
- Isolate the variable by subtracting 6.
- Final answers are \( x = -6 + 5\sqrt{2} \) and \( x = -6 - 5\sqrt{2} \).
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