Question

If x = (2 -√3). Find the value of x^2 + 1/x^2.

Answer 1

x = (2 - √3)

1/x = 1/(2 - √3)

= (2 + √3)/(2 - √3)(2 + √3)

= (2 + √3)/(2² - √3²)

=(2 + √3)/(4 -3) = (2 + √3)

hence , 1/x = (2 + √3)

so, x + 1/x = (2 - √3) + (2 + √3) = 4

now,
x² + 1/x² = (x + 1/x )² -2

=( 4)² -2 = 14 (answer )

Answer 2

x=(2-√3)
1/x= 1/(2-√3)
=(1×2+√3)/(2-√3)(2+√3). (rationalizing the denominator)
=(2+√3)/{2²-(√3)²}
=(2+√3)/(4-3)
=(2+√3)/1 = 2+√3
x= 2-√3 , 1/x=2+√3
(x+1/x)²= x²+(1/x)²+2×x×1/x

(2-√3+2+√3)²= x²+(1/x)²+2
4²=x²+(1/x)²+2
16=x²+(1/x)²+2
16-2=x²+(1/x)²
14=x²+(1/x)²
hope this helps