Question

If x= 3+√8, find x²+(1/x²)​

Answer 1

Hope it is helpful !!!!

Step-by-step explanation:

x = 3+ √8

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)Or we can write 1/x = (3-√8)

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)Or we can write 1/x = (3-√8)Put the value and find out

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)Or we can write 1/x = (3-√8)Put the value and find outx² +(1/x²) = x² + (1/x)²

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)Or we can write 1/x = (3-√8)Put the value and find outx² +(1/x²) = x² + (1/x)²So x² + 1/ x² = (3 +√8)² + (3 -√8)²

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)Or we can write 1/x = (3-√8)Put the value and find outx² +(1/x²) = x² + (1/x)²So x² + 1/ x² = (3 +√8)² + (3 -√8)²= 9+6√8+8 + 9–6√8+8

x = 3+ √8x = (3+√8)(3-√8) / (3-√8)x = (9 -8) / (3-√8)x = 1/ (3 -√8)Or we can write 1/x = (3-√8)Put the value and find outx² +(1/x²) = x² + (1/x)²So x² + 1/ x² = (3 +√8)² + (3 -√8)²= 9+6√8+8 + 9–6√8+8= 34. Ans…………...

Answer 2

$x = 3 + \sqrt{8}$

Squaring both sides, we have

 ${x}^{2} = ( {3 + \sqrt{8} })^{2}$

$\longrightarrow{\green{}}$ $({3})^{2} + ({ \sqrt{8} })^{2} + 2 \times 3 \times \sqrt{8}$

$\longrightarrow{\green{}}$ $9 + 8 + 12 \sqrt{2}$

$\longrightarrow{\green{}}$ $17 + 12 \sqrt{2}$

Let us solve for $\frac{1}{ {x}^{2} }$ by rationalising the denominator.

$\longrightarrow{\green{}}$ $\frac{1}{17 + 12 \sqrt{2} } \times \frac{17 - 12 \sqrt{2} }{17 - 12 \sqrt{2} }$

$\longrightarrow{\green{}}$ $\frac{17 - 12 \sqrt{2} }{ ({17})^{2} - ({12 \sqrt{2} })^{2} }$

$\longrightarrow{\green{}}$ $\frac{17 - 12 \sqrt{2} }{289 - 288}$

$\longrightarrow{\green{}}$ $\frac{17 - 12 \sqrt{2} }{1}$

$\longrightarrow{\green{}}$ $17 - 12 \sqrt{2}$

And finally substitute their values in the expression below.

$\longrightarrow{\green{}}$ ${x}^{2} + ( { \frac{1}{x} })^{2}$

$\longrightarrow{\green{}}$ $17 + 12 \sqrt{2} + 17 - 12 \sqrt{2}$

$\longrightarrow{\green{34}}$