Question

if x= 3+2√2 FIND THE VALUE X^2+1/X^2​

Answer 1

Answer: 34

Step-by-step explanation:

x = 3 + 2√2 .......(eq. 1)

1/x = 1/( 3 + 2√2 )

= ( 3 - 2√2 )/[ ( 3 + 2√2 )( 3 - 2√2 )]

= ( 3 - 2√2 )/ [ 3² - ( 2√2 )² ]

= ( 3 - 2√2 ) / ( 9 - 8 )

= 3 - 2√2 ......(eq. 2 )

x + 1/x = 3 + 2√2 + 3 - 2√2

= 6 ...... (eq.3)

Now,

Squaring eq. (3), we get:

( x + 1/x )² = 6²

x² + 1/x² + 2 = 36

x² + 1/x² = 36 - 2

x² + 1/x² = 34

Answer 2

Question:

If x = 3 + 2√2, find the value of x² + 1/x².

Solution:

ATQ, We have to find x² + 1/x², and we've been given that x = 3 + 2√2. Therefore,

$\Longrightarrow \sf x = 3 + 2\sqrt{2}\\ \\ \\\sf Taking \ reciprocal \ we \ get,\\ \\ \\\Longrightarrow \sf \dfrac{1}{x} = \dfrac{1}{3 + 2\sqrt{2}} \\ \\ \\\Longrightarrow \sf \dfrac{1}{x} = \dfrac{1}{3 + 2\sqrt{2}} \times \dfrac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} \\ \\ \\\sf Using \ (a + b) \ (a - b) = a^2 - b^2 \ we \ get,\\ \\ \\\Longrightarrow \sf \dfrac{1}{x} = \dfrac{3 - 2\sqrt{2}}{\left(3\right)^2 - \left(2\sqrt{2}\right)^2}\\ \\ \\\Longrightarrow \sf \dfrac{1}{x} = \dfrac{3 - \sqrt{2}}{9 - 8}$

$\Longrightarrow \sf \dfrac{1}{x} = \dfrac{3 - \sqrt{2}}{1}\\ \\ \\\Longrightarrow \sf \dfrac{1}{x} =3 - \sqrt{2} \ \ \longmapsto \textcircled{\sf \scriptsize 1}$

Now, Let's find the value of x + 1/x.

Substitute the value of 'x', and '1/x' from the question and from above respectively.

$\Longrightarrow \sf x + \dfrac{1}{x} = 3 \ + \ 2\sqrt{2} \ + \ 3 \ - \ 2\sqrt{3}\\ \\ \\ \Longrightarrow \sf x + \dfrac{1}{x} = 3 + 3\\ \\ \\\Longrightarrow \sf x + \dfrac{1}{x} = 6\\ \\ \\\sf Squaring \ on \ both \ sides \ we \ get,\\ \\ \\\Longrightarrow \sf \left(x + \dfrac{1}{x} \right)^2 = \left(6\right)^2\\ \\ \\ \\\sf Using \ the \ identity \ (a + b)^2 = a^2 + b^2 + 2ab \ we \ get,\\ \\ \\\Longrightarrow \sf x^2 + \dfrac{1}{x^2} + 2\left( x\right) \left(\dfrac{1}{x}\right) = 36$

$\Longrightarrow \sf x^2 + \dfrac{1}{x^2} + 2 = 36 \\ \\ \\ \\\Longrightarrow \sf x^2 + \dfrac{1}{x^2} = 36 - 2 \\ \\ \\ \\\Longrightarrow \sf x^2 + \dfrac{1}{x^2} = 34$

∴ Answer will be 34.