Question

x^2=5+2√6
Find the value of x​

Answer 1

x² = 5+2√6

then ,

x = √(5+2√6) = √5 + 6√2

⇒ Value of x is √5 + 6√2

Answer 2

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: {x}^{2} = 5 + 2 \sqrt{6}$

$\large\underline{\sf{To\:Find - }}$

The value of x.

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\: {x}^{2} = 5 + 2 \sqrt{6}$

can be rewritten as

$\rm :\longmapsto\: {x}^{2} = 3 + 2 + 2 \sqrt{6}$

can be again rewritten as

$\rm :\longmapsto\: {x}^{2} = {( \sqrt{3} )}^{2} + {( \sqrt{2} )}^{2} + 2 \sqrt{3 \times 2}$

can be rewritten as

$\rm :\longmapsto\: {x}^{2} = {( \sqrt{3} )}^{2} + {( \sqrt{2} )}^{2} + 2 \sqrt{3} \sqrt{2}$

We know,

$\boxed{ \rm \: {x}^{2} + {y}^{2} + 2xy = {(x + y)}^{2} }$

So, using this identity, we get

$\rm :\longmapsto\: {x}^{2} = {( \: \sqrt{3} + \sqrt{2} \: ) }^{2}$

$\bf\implies \:x \: = \: \pm \: ( \: \sqrt{3} + \sqrt{2} \: )$

Additional Information :-

Let's solve the same type of problem!!!

Question :-

$\rm :\longmapsto\:If \: {x}^{2} = 3 - 2 \sqrt{2}, \: find \: the \: value \: of \: x$

Answer :-

Given that

$\rm :\longmapsto\: {x}^{2} = 3 - 2 \sqrt{2}$

$\rm :\longmapsto\: {x}^{2} = 2 + 1 - 2 \sqrt{2}$

$\rm :\longmapsto\: {x}^{2} = {( \sqrt{2} )}^{2} + {(1)}^{2} - 2 \sqrt{2 \times 1}$

$\rm :\longmapsto\: {x}^{2} = {( \sqrt{2} )}^{2} + {(1)}^{2} - 2 \sqrt{2} \times 1$

We know,

$\boxed{ \rm \: {x}^{2} + {y}^{2} - 2xy = {(x - y)}^{2} }$

So, using this identity, we get

$\rm :\longmapsto\: {x}^{2} = {( \sqrt{2} - 1)}^{2}$

$\bf\implies \:x \: = \: \pm \: ( \: \sqrt{2} - 1 \: )$