To rewrite the equation \(x^{2} + 7x + 12 = 0\) by completing the square, we follow these steps:
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Start with the original equation: \[ x^2 + 7x + 12 = 0 \]
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Move the constant term to the other side: \[ x^2 + 7x = -12 \]
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To complete the square, take the coefficient of \(x\) (which is 7), divide it by 2, and square it: \[ \left(\frac{7}{2}\right)^2 = \frac{49}{4} \]
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Add \(\frac{49}{4}\) to both sides of the equation: \[ x^2 + 7x + \frac{49}{4} = -12 + \frac{49}{4} \]
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Simplify the right side. First, convert \(-12\) into a fraction with a denominator of 4: \[ -12 = -\frac{48}{4} \] So, \[ -12 + \frac{49}{4} = -\frac{48}{4} + \frac{49}{4} = \frac{1}{4} \]
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The left-hand side can be factored since we have completed the square: \[ (x + \frac{7}{2})^2 = \frac{1}{4} \]
Thus, the equation by completing the square is: \[ (x + \frac{7}{2})^2 = \frac{1}{4} \]
So, you can fill in the blanks as follows: \[ \begin{align*} x + \frac{7}{2} \ \frac{1}{4} \end{align*} \]
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