To simplify the expression [tex]\(\frac{\sqrt{3}}{3 - 2 \sqrt{x}}\)[/tex], follow these steps:
1. Identify the Conjugate:
The given expression has a radical term in the denominator. To simplify, we'll multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 - 2 \sqrt{x}\)[/tex] is [tex]\(3 + 2 \sqrt{x}\)[/tex].
2. Multiply the Numerator and Denominator by the Conjugate:
[tex]\[
\frac{\sqrt{3}}{3 - 2 \sqrt{x}} \times \frac{3 + 2 \sqrt{x}}{3 + 2 \sqrt{x}} = \frac{\sqrt{3} (3 + 2 \sqrt{x})}{(3 - 2 \sqrt{x})(3 + 2 \sqrt{x})}
\][/tex]
3. Simplify the Denominator:
The denominator is a difference of squares:
[tex]\[
(3 - 2 \sqrt{x})(3 + 2 \sqrt{x}) = 3^2 - (2 \sqrt{x})^2 = 9 - 4x
\][/tex]
4. Expand the Numerator:
Distribute [tex]\(\sqrt{3}\)[/tex] across the terms in the numerator:
[tex]\[
\sqrt{3} (3 + 2 \sqrt{x}) = 3 \sqrt{3} + 2 \sqrt{3} \sqrt{x}
\][/tex]
However, we can represent the original denominator [tex]\(3 - 2 \sqrt{x}\)[/tex] in a negative form:
[tex]\[
3 - 2 \sqrt{x} = -(2 \sqrt{x} - 3)
\][/tex]
Thus, rewriting the expression:
[tex]\[
\frac{\sqrt{3}}{3 - 2 \sqrt{x}} = \frac{\sqrt{3}}{-(2 \sqrt{x} - 3)} = -\frac{\sqrt{3}}{2 \sqrt{x} - 3}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
-\frac{\sqrt{3}}{2 \sqrt{x} - 3}
\][/tex]
