Simplify: $$ \frac{3-\sqrt{5}}{2+\sqrt{5}} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{3 - \sqrt{5}}{2 + \sqrt{5}}
$$

we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is $$2 - \sqrt{5}$$:

$$
\frac{3 - \sqrt{5}}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{(3 - \sqrt{5})(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}
$$

**Step 1: Expand the Numerator**

$$
(3 - \sqrt{5})(2 - \sqrt{5}) = 3 \cdot 2 + 3 \cdot (-\sqrt{5}) + (-\sqrt{5}) \cdot 2 + (-\sqrt{5}) \cdot (-\sqrt{5})
$$
$$
= 6 - 3\sqrt{5} - 2\sqrt{5} + 5
$$
$$
= 11 - 5\sqrt{5}
$$

**Step 2: Simplify the Denominator**

$$
(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
$$

**Step 3: Combine the Results**

$$
\frac{11 - 5\sqrt{5}}{-1} = -11 + 5\sqrt{5} = 5\sqrt{5} - 11
$$

**Final Answer:**

$$
5\sqrt{5} - 11
$$
$$ 5\sqrt{5} - 11 $$

Simplify:

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