Question

Find the roots of the equation \( x + \frac{1}{x} = 3, \quad x \neq 0 \).

Hint:

Convert equations into standard quadratic form before solving.

Answer 1

Multiplying both sides by \( x \):
\[ x^2 + 1 = 3x. \] \[ x^2 - 3x + 1 = 0. \] Using the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(1)}}{2(1)}. \] \[ = \frac{3 \pm \sqrt{9 - 4}}{2}. \] \[ = \frac{3 \pm \sqrt{5}}{2}. \] Thus, the roots are:
\[ \frac{3+\sqrt{5}}{2}, \quad \frac{3-\sqrt{5}}{2}. \]