Rationalize $\frac{2 - \sqrt5}{3 - \sqrt5}$

Analysis

To rationalize a fraction with a radical (surd) in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(3 - \sqrt{5})$ is $(3 + \sqrt{5})$.

Concept

The goal is to eliminate the square root from the denominator. We use the difference of two squares identity: $(a - b)(a + b) = a^2 - b^2$

Step-by-Step Solution

Step 1: Set up the multiplication

Step 2: Expand the numerator $(2 - \sqrt{5})(3 + \sqrt{5})$ $= 2(3) + 2(\sqrt{5}) - 3(\sqrt{5}) - (\sqrt{5})(\sqrt{5})$ $= 6 + 2\sqrt{5} - 3\sqrt{5} - 5$ $= (6 - 5) + (2\sqrt{5} - 3\sqrt{5})$ $= 1 - \sqrt{5}$

Step 3: Expand the denominator $(3 - \sqrt{5})(3 + \sqrt{5})$ $= 3^2 - (\sqrt{5})^2$ $= 9 - 5$ $= 4$

Step 4: Combine the results

Conclusion

Comparing our result to the given options, we can see that it matches option B.

Exam Tip: When rationalizing, always remember to change the sign in the middle of the denominator to get the conjugate. If the denominator is $(a - \sqrt{b})$, the conjugate is $(a + \sqrt{b})$. If it is $(a + \sqrt{b})$, the conjugate is $(a - \sqrt{b})$. This simple sign flip is the key to clearing the surd!