Question

x equal to 2 + root 3 then find the value of x square + 1 upon x square

Answer 1

Hence,


X² + 1/X² = 14




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Answer 2

Answer:

$x^{2}+\frac{1}{x^{2}}$

=$14$

Explanation:

Given $x = 2+\sqrt{3}$---(1)

i ) $\frac{1}{x}$

=$\frac{1}{2+\sqrt{3}}$

Rationalising the denominator, we get,

= $\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}$

=$\frac{2-\sqrt{3}}{2^{2}-(\sqrt{3})^{2}}$

/* By algebraic identity:

(a+b)(a-b)= a²-b² */

= $\frac{2-\sqrt{3}}{(4-3)}$

= $2-\sqrt{3}$---(2)

ii) $x+\frac{1}{x}$

= =$2+\sqrt{3}+2-\sqrt{3}$

= $4$ ----(3) /* From (1)&(2} */

Now ,

$x^{2}+\frac{1}{x^{2}}$

=$\left(x+\frac{1}{x}\right)^{2}-2$

/* By algebraic identity:

a²+b² = (a+b)²-2ab */

= $4^{2}-2$ /* From (3) */

=$ 16-2$

= $14$

Therefore,

$x^{2}+\frac{1}{x^{2}}$

=$14$

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