if \(x=\frac{\sqrt{\sqrt{5}+1}}{\sqrt{\sqrt{5}-1}}\) then the value of \(5 x^2-5 x-1\) will be.

Solution

Given:
\[ x = \frac{\sqrt{\sqrt{5} + 1}}{\sqrt{\sqrt{5} – 1}} \]

We need to find the value of \(5x^2 – 5x – 1\).

Step 1: Simplify \(x\)

Multiply the numerator and denominator by \(\sqrt{\sqrt{5} + 1}\):
\[ x = \sqrt{\frac{(\sqrt{5} + 1)(\sqrt{5} + 1)}{(\sqrt{5} – 1)(\sqrt{5} + 1)}} = \sqrt{\frac{(\sqrt{5} + 1)^2}{5 – 1}} = \frac{\sqrt{5} + 1}{2} \]

Step 2: Substitute \(x\) into \(5x^2 – 5x – 1\)

\[ 5x^2 – 5x – 1 = 5\left(\frac{\sqrt{5} + 1}{2}\right)^2 – 5\left(\frac{\sqrt{5} + 1}{2}\right) – 1 \]

Simplify the expression:
\[ = 5 \times \frac{(3 + \sqrt{5})}{2} – \frac{5\sqrt{5} – 5 – 2}{2} \]
\[ = \frac{15 + 5\sqrt{5} – 5\sqrt{5} – 7}{2} \]
\[ = \frac{8}{2} \]
\[ = 4 \]

Conclusion

The value of \(5x^2 – 5x – 1\) is 4.