If x+\frac{1}{x}=2 \sqrt{5}, then what is the value of {(x^{4}+{1}/{x^{2}})}/{x^{2}+1} ?

SSC CGL 23 Aug 2021 Shift 2 Paper

Question : 69 of 100

Marks: +1, -0

x+\;{1}/{x}=2 √{5}

, then what is the value of

\;{(x^{4}+{1}/{x^{2}})}/{x^{2}+1} ?

Solution:

Given,

x+\;{1}/{x}=2 √{5}

⇒ \;\; (x^{2}+1)=2 √{5} x

Now,

\;{(x^{4}+{1}/{x^{2}})}/{(x^{2}+1)}=\;{(x^{8}+1)}/{x^{2}(x^{2}+1)}

We know that,

(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)

Here,

a

x^{2}, b=1

(x^{2}+1)^{3}=x^{8}+1+3 × x^{2} × 1(x^{2}+1)

⇒(2 √{5} x)^{3}=x^{8}+1+3 x^{2}(2 √{5} x)

[ ∵ x^{2}+1=2 √{5} x]

⇒ \;\; 40 √{5} x^{3}=x^{8}+1+6 √{5} x^{3}

⇒ \;\; (x^{8}+1)=40 √{5} x^{3}-6 √{5} x^{3}

=34 √{5} x^{3}

∴ \;{(x^{8}+1)}/{x^{2}(x^{2}+1)}=\;{34 √{5} x^{3}}/{x^{2} × 2 √{5} x}

=\;{34 √{5} x^{3}}/{2 √{5} x^{3}}=17

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