Question

if a is equals to 3 + root 5 upon 2 then find the value of a square + 1 upon a square

Answer 1

$a = 3 + \frac{ \sqrt{5} }{2}$
${a}^{2} = 9 + \frac{5}{4} + 3 \sqrt{5}$
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Answer 2

Answer:

Value of given expression is 7.

Step-by-step explanation:

Given:

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

To find: $a^2+\frac{1}{a^2}$

$a^2=(\frac{3+\sqrt{5}}{2})^2=\frac{(3+\sqrt{5})^2}{2^2}=\frac{3^2+(\sqrt{5}^2)+2\times3\times\sqrt{5}}{4}=\frac{9+5+6\sqrt{5}}{4}=\frac{14+6\sqrt{5}}{4}=\frac{7+3\sqrt{5}}{2}$

Now,

$\frac{1}{a^2}=\frac{2}{7+3\sqrt{5}}=\frac{2}{7+3\sqrt{5}}\times\frac{7-3\sqrt{5}}{7-3\sqrt{5}}=\frac{2(7-3\sqrt{5})}{(7+3\sqrt{5})(7-3\sqrt{5})}=\frac{14-6\sqrt{5})}{49+9\times5}=\frac{14-6\sqrt{5}}{4}=\frac{7-3\sqrt{5}}{2}$

Consider,

$a^2+\frac{1}{a^2}=\frac{7+3\sqrt{5}}{2}+\frac{7-3\sqrt{5}}{2}=\frac{7+3\sqrt{5}+7-3\sqrt{5}}{2}=\frac{14}{2}=7$

Therefore, Value of given expression is 7.