Question

If x=(5-√21÷2) then find x+1/x

Answer 1

$\textbf{Given:}$

$x=\displaystyle\frac{5-\sqrt{21}}{2}$

$\implies\displaystyle\frac{1}{x}=\frac{2}{5-\sqrt{21}}$

$=\displaystyle\frac{2}{5-\sqrt{21}}\times\frac{5+\sqrt{21}}{5+\sqrt{21}}$

$=\displaystyle\frac{2(5+\sqrt{21})}{25-21}$

$=\displaystyle\frac{2(5+\sqrt{2})}{4}$

$=\displaystyle\frac{5+\sqrt{2}}{2}$

$\text{Now,}$

$\displaystyle\;x+\frac{1}{x}$

$=\displaystyle\frac{5-\sqrt{21}}{2}+\frac{5+\sqrt{21}}{2}$

$=\displaystyle\frac{5+5}{2}$

$=5$

$\therefore\boxed{\bf\,x+\frac{1}{x}=5}$

Answer 2

Step-by-step explanation:

x=

2

5−

21

\implies\displaystyle\frac{1}{x}=\frac{2}{5-\sqrt{21}}⟹

x

1

=

5−

21

2

=\displaystyle\frac{2}{5-\sqrt{21}}\times\frac{5+\sqrt{21}}{5+\sqrt{21}}=

5−

21

2

×

5+

21

5+

21

=\displaystyle\frac{2(5+\sqrt{21})}{25-21}=

25−21

2(5+

21

)

=\displaystyle\frac{2(5+\sqrt{2})}{4}=

4

2(5+

2

)

=\displaystyle\frac{5+\sqrt{2}}{2}=

2

5+

2

\text{Now,}Now,

\displaystyle\;x+\frac{1}{x}x+

x

1

=\displaystyle\frac{5-\sqrt{21}}{2}+\frac{5+\sqrt{21}}{2}=

2

5−

21

+

2

5+

21

=\displaystyle\frac{5+5}{2}=

2

5+5

=5=5

\therefore\boxed{\bf\,x+\frac{1}{x}=5}∴

x+

x

1

=5