Question

If x=2+√3, find the value of x²+1/x²

Answer 1

x²+1/x² =
`(x + 1/x)^(2)` - 2 * x * ( 1 / x) = ( 2 + √3 + 2 - √3)^2 - 2 = 16 - 2 = 14.

Answer 2

`x^2+ (1)/(x^2) =x^2+2\cdot x\cdot (1)/(x) +((1)/(x))^2-2\cdot x\cdot (1)/(x) =(x+(1)/(x))^2-2\\\\x=2+ √(3) \ \ \ \Rightarrow\ \ \ x+ (1)/(x) =2+ √(3)+ (1)/(2+ √(3)) =2+ √(3)+ (2- √(3) )/((2+ √(3))(2- √(3))) =\\\\=2+ √(3) + (2- √(3) )/(4-3) =2+ √(3) +2- √(3) =4\\\\x+ (1)/(x) =4\ \ \ \Rightarrow\ \ \ (x+(1)/(x))^2-2=4^2-2=16-2=14`