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  • If a=3+sqroot 5 by 2, then find the value of a power 2 + 1 by a power 2

If  a= \frac{3+\sqrt{5}}{2} then find the value of a^{2}= \frac{1}{a^{2}}

Answers (1)

Answer. 7
Solution.Given that :- a= \frac{3+\sqrt{5}}{2}

\therefore \frac{1}{a}= \frac{2}{3+\sqrt{5}}
On rationalizing the denominator, we get

\frac{1}{a}= \frac{2\left ( 3-\sqrt{5} \right )}{\left ( 3+\sqrt{5} \right )\left ( 3-\sqrt{5} \right )}
Using   (a – b) (a + b) = a2 – b2

= \frac{6-2\sqrt{5}}{3^{2}-\sqrt{5}^{2}}                

= \frac{6-2\sqrt{5}}{9-5}

= \frac{6-2\sqrt{5}}{4}
= \frac{3-\sqrt{5}}{2}
Also, \left ( a+\frac{1}{a} \right )^{2}= a^{2}+\frac{1}{a^{2}}+2
Substituting the values of a and \frac{1}{a}
We get, \left ( \frac{3+\sqrt{5}}{2}+\frac{3-\sqrt{5}}{2} \right )^{2}= \left ( a^{2}+\frac{1}{a^{2}}+2 \right )
\therefore a^{2}+\frac{1}{a^{2}}+2= \left ( \frac{3+\sqrt{5}+3-\sqrt{5}}{2} \right )^{2}
= (3)2 = 9

\therefore a^{2}+\frac{1}{a^{2}}= 9-2= 7
Hence the correct answer is 7.

Posted by

infoexpert27

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