To complete the square for the equation \( x^2 + 3x = 6 \), we need to add a number to both sides of the equation.
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Start with the left-hand side: \( x^2 + 3x \).
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To complete the square, take the coefficient of \( x \) (which is 3), divide it by 2, and then square the result: \[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]
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Add \( \frac{9}{4} \) to both sides of the equation: \[ x^2 + 3x + \frac{9}{4} = 6 + \frac{9}{4} \]
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Now, to simplify the right-hand side, convert 6 to a fraction with a denominator of 4: \[ 6 = \frac{24}{4} \] So, \[ 6 + \frac{9}{4} = \frac{24}{4} + \frac{9}{4} = \frac{33}{4} \]
Now the equation will look like: \[ x^2 + 3x + \frac{9}{4} = \frac{33}{4} \]
Thus, the number that should be added to both sides of the equation to complete the square is \( \frac{9}{4} \).