If \(x = 8 - 2\sqrt {15}\), then find the value of \({\left( {\frac{{{\rm{x}} + {\rm{}}1}}{{\sqrt {\rm{x}} }}} \right)^2}\)

Question

Question

Answer 1

Given:

x = 8 – 2√(15)

Calculation:

x = (√5)² + (√3)² – 2√(15)

⇒ x = (√5 - √3)²

⇒ √x = √5 - √3

And 1/√x = (√5 + √3)/2

According to question,

\({\left( {\sqrt x + \frac{1}{{\sqrt x }}} \right)^2} = {\left[ {\sqrt 5 - \sqrt 3 + \left( {\frac{{\sqrt 5 + \sqrt 3 \;}}{2}} \right)} \right]^2}\)

\(\Rightarrow {\left[ {\frac{{3\sqrt 5 - \sqrt 3 }}{2}} \right]^2}\)

\(\Rightarrow \left[ {\frac{{45 + 3 - 6\sqrt {15} }}{4}} \right]\)

\(\Rightarrow \frac{{48 - 6\sqrt {15} }}{4}\)

⇒ \(12 - \frac{{3\sqrt {15} }}{2}\)