Question

For the equation ${(x+\frac{1}{x})}^2-5(x+\frac{1}{x})+6=0,$

• A. has all roots integers
• B. has all roots irrational
• C. two roots are rational, two roots are irrational
• D. two roots are real, two rootsare imaginary

Answer 1

The correct option is C

two roots are rational, two roots are irrational

Let $x+\frac{1}{x}=y$

Given equation is $y^2-5y+6=0$

(y - 2) (y - 3) = 0

y = 2, y =3

$\begin{array}{c}{x}^2-2x+1=0\\ x=\frac{2}{3},\frac{3}{2}\end{array}|\begin{array}{c}{x}^2-3x+1=0\\ x=\frac{3\pm \sqrt{9-4}}{2}=\frac{3\pm \sqrt{5}}{2}\end{array}$

Equation has two roots rational two roots irrational.