To solve the equation \( x^2 + 12x + 26 = 0 \) by completing the square, follow these steps:
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Rearrange the terms: \[ x^2 + 12x = -26 \]
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Complete the square:
- Take the coefficient of \( x \) (which is 12), divide it by 2 to get 6, and then square it to get \( 6^2 = 36 \).
- Add 36 to both sides of the equation: \[ x^2 + 12x + 36 = -26 + 36 \] Simplifying the right side: \[ x^2 + 12x + 36 = 10 \]
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Rewrite the left side as a square: \[ (x + 6)^2 = 10 \]
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Take the square root of both sides: \[ x + 6 = \pm \sqrt{10} \]
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Isolate \( x \):
- For the positive case: \[ x = -6 + \sqrt{10} \]
- For the negative case: \[ x = -6 - \sqrt{10} \]
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Final solutions: \[ x = -6 + \sqrt{10} \quad \text{and} \quad x = -6 - \sqrt{10} \]
Therefore, the solutions to the equation \( x^2 + 12x + 26 = 0 \) are: \[ x = -6 + \sqrt{10} \quad \text{and} \quad x = -6 - \sqrt{10} \]