Let's fill in the steps to complete the square for the given 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 other side: \[ x^2 + 12x = 14 \]
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To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is 12, half of it is 6, and squaring it gives us 36: \[ x^2 + 12x + 36 = 14 + 36 \]
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This simplifies to: \[ (x + 6)^2 = 50 \]
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Now, to solve for \(x\), we need to take the square root of both sides: \[ \sqrt{(x + 6)^2} = \sqrt{50} \]
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Which gives us: \[ x + 6 = \pm \sqrt{50} \]
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Now, isolate the variable \(x\) by subtracting 6 from both sides: \[ x = -6 \pm \sqrt{50} \]
Thus, the detailed breakdown yields the following answers:
Final answers are:
- \( x = -6 + \sqrt{50} \)
- \( x = -6 - \sqrt{50} \)
In terms of your fill-in-the-blanks, the completed parts would be:
- \( x^2 + 12x + 36 \)
- \( 14 + 36 \)
- \( (x + 6)^2 \)
- Take the square root of both sides of the equation
- Which gives \( x = -6 \pm \sqrt{50} \)
- Isolate the variable by subtracting 6 from both sides.
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