To simplify the expression \( x = \frac{9 \pm 2\sqrt{3}}{2} \), we can separate the two cases for the plus and minus:
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For the case with the plus sign: \[ x_1 = \frac{9 + 2\sqrt{3}}{2} \]
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For the case with the minus sign: \[ x_2 = \frac{9 - 2\sqrt{3}}{2} \]
Now, we can simplify each case further:
Case 1: \( x_1 = \frac{9 + 2\sqrt{3}}{2} \)
\[ x_1 = \frac{9}{2} + \frac{2\sqrt{3}}{2} = \frac{9}{2} + \sqrt{3} \]
Case 2: \( x_2 = \frac{9 - 2\sqrt{3}}{2} \)
\[ x_2 = \frac{9}{2} - \frac{2\sqrt{3}}{2} = \frac{9}{2} - \sqrt{3} \]
Thus, the simplified versions of \( x \) are: \[ x_1 = \frac{9}{2} + \sqrt{3} \quad \text{and} \quad x_2 = \frac{9}{2} - \sqrt{3} \]
So, the final simplified form of the expression \( x \) can be written as: \[ x = \frac{9}{2} \pm \sqrt{3} \]