Question

if a = 3+√5/2 then find the value of a^2+1/a^2​

Answer 1

Answer:

$a^2+\frac{1}{a^2}=7$

$Refer \: \: to \: \: steps \: \: for \: \: attachment...$

Answer 2

Given,

$\: \: \: \: \: \: \mathtt{a \: = 3 + \frac{ \sqrt{5} }{2} }$

Now,

$\mathtt{ {a}^{2} + \frac{1}{ {a}^{2} } }$

$\mathtt{ = (a + \frac{1}{a} ) ^{2} } - 2$

$\large = (3 + \frac{ \sqrt{5} }{2} + \frac{1}{3 + \frac{ \sqrt{5} }{2} } )^{2} - 2$

$\large = (\frac{6 + \sqrt{5} }{2} + \frac{1}{ \frac{6 + \sqrt{5} }{2}} ) ^{2} - 2$

$= (\frac{6 + \sqrt{5} }{2} + \frac{2}{6 + \sqrt{5} })^{2} - 2$

$= ( \frac{(6 + \sqrt{5} )^{2} - 4 }{2(6 + \sqrt{5} )} )^{2} - 2$

$= (\frac{41 + 12 \sqrt{5} }{12 + 2 \sqrt{5} } )^{2} - 2$

$= \frac{984 \sqrt{5} + 2401}{48 \sqrt{5} + 164} - 2$

$= \large{ \boxed{ \mathtt{ \color{blue}{\bold{ 3 \sqrt{5} + \frac{33}{4} }}}}}$