To rationalize the denominator and simplify the expression [tex]\(\frac{-5}{3 \sqrt{5} + 2}\)[/tex], follow these detailed steps:
1. Identify the expression:
[tex]\[
\frac{-5}{3 \sqrt{5} + 2}
\][/tex]
2. Multiply the numerator and the denominator by the conjugate of the denominator:
The conjugate of [tex]\(3 \sqrt{5} + 2\)[/tex] is [tex]\(3 \sqrt{5} - 2\)[/tex]. Multiplying by this conjugate allows us to eliminate the square root in the denominator.
[tex]\[
\frac{-5 \cdot (3 \sqrt{5} - 2)}{(3 \sqrt{5} + 2) \cdot (3 \sqrt{5} - 2)}
\][/tex]
3. Expand the numerator:
[tex]\[
-5 \cdot (3 \sqrt{5} - 2) = -5 \cdot 3 \sqrt{5} + (-5) \cdot (-2) = -15 \sqrt{5} + 10
\][/tex]
4. Expand the denominator:
Using the difference of squares formula [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex], where [tex]\(a = 3 \sqrt{5}\)[/tex] and [tex]\(b = 2\)[/tex]:
[tex]\[
(3 \sqrt{5} + 2)(3 \sqrt{5} - 2) = (3 \sqrt{5})^2 - (2)^2
\][/tex]
5. Simplify the denominator:
Evaluate the squares:
[tex]\[
(3 \sqrt{5})^2 = 9 \cdot 5 = 45
\][/tex]
[tex]\[
2^2 = 4
\][/tex]
Thus:
[tex]\[
(3 \sqrt{5})^2 - 2^2 = 45 - 4 = 41
\][/tex]
6. Combine the results:
The expression now becomes:
[tex]\[
\frac{-15 \sqrt{5} + 10}{41}
\][/tex]
So, the rationalized and simplified form of the expression [tex]\(\frac{-5}{3 \sqrt{5} + 2}\)[/tex] is:
[tex]\[
\frac{-15 \sqrt{5} + 10}{41}
\][/tex]
