Question

if x=root 6+root 5 ,then the value of x^2+1/x^2
(a) 2 root 6
(b) 2 root 5
(c) 24
(d) 22​

Answer 1

Answer:

The answer is d.) 22

Step-by-step explanation:

It is given that x = $\sqrt{6} + \sqrt{5}$

We have to find the value of $x^2 + \frac{1}{x^2}$

Substituting the value of x in the equation given above we get

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

= $(6 + 5 + 2\sqrt{30}) + \frac{1}{(6 + 5 + 2\sqrt{30})}$

= $(11 + 2\sqrt{30} ) + \frac{1}{11 + 2\sqrt{30} }$      ....   (i)

We can simplify the expression by multiplying the numerator and denominator of the second term by the conjugate of its denominator which is $11 - 2\sqrt{30}$.

The expression in (i) simplifies to

${(11 + 2\sqrt{30}) + \frac{(11 - 2\sqrt{30})}{(11 + 2\sqrt{30})(11 - 2\sqrt{30})}$

$= (11 + 2\sqrt{30}) + \frac{(11 - 2\sqrt{30})}{121 -120}$     ..... (ii) we get this from the algebraic formula that          

                                                       (a + b)(a - b)  = $a^2$ - $b^2$

The expression in (ii)  simplifies to

$(11 + 2\sqrt{30}) + (11 - 2\sqrt{30})$

= 22

The answer is d.) 22