Question
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Tutor-Verified Answer
Answer
\[
-\frac{\sqrt{5}+5}{10}
\]
Solution
We start with the expression
\[
\frac{2}{\sqrt{5}-5}.
\]
**Step 1: Multiply by the conjugate**
Multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{5}+5\):
\[
\frac{2}{\sqrt{5}-5} \cdot \frac{\sqrt{5}+5}{\sqrt{5}+5}.
\]
**Step 2: Simplify the expression**
The numerator becomes
\[
2(\sqrt{5}+5).
\]
The denominator is the difference of two squares:
\[
(\sqrt{5}-5)(\sqrt{5}+5) = (\sqrt{5})^2 - 5^2 = 5 - 25 = -20.
\]
Thus, the expression becomes:
\[
\frac{2(\sqrt{5}+5)}{-20}.
\]
**Step 3: Reduce the fraction**
Factor \(2\) out of the numerator and simplify with \(-20\):
\[
\frac{2(\sqrt{5}+5)}{-20} = \frac{\sqrt{5}+5}{-10} = -\frac{\sqrt{5}+5}{10}.
\]
The simplified form is
\[
-\frac{\sqrt{5}+5}{10}.
\]
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Simplify this solution Mind Expander
To simplify \( \frac{2}{\sqrt{5}-5} \), we can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{5}+5 \). This gives us: \[ \frac{2(\sqrt{5}+5)}{(\sqrt{5}-5)(\sqrt{5}+5)} = \frac{2(\sqrt{5}+5)}{5 - 25} = \frac{2(\sqrt{5}+5)}{-20} = -\frac{\sqrt{5}+5}{10} \] So, the simplified form is \( -\frac{\sqrt{5}+5}{10} \). To double-check, remember the property of conjugates: \( (a - b)(a + b) = a^2 - b^2 \), which in our case confirms that the denominator becomes \( 5 - 25 = -20 \). If you ever find a complex denominator, multiplying by its conjugate is a nifty trick to make it manageable!
