Question

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

Hint:

For quadratic equations, the quadratic formula is a powerful tool to find the roots.

Answer 1

We are given the equation \[ x + \frac{1}{x} = 3, \quad x \neq 0. \] Multiply both sides by \( x \) (since \( x \neq 0 \)) to eliminate the fraction: \[ x^2 + 1 = 3x. \] Now, rearrange the terms to form a quadratic equation: \[ x^2 - 3x + 1 = 0. \] We can solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 1 \), \( b = -3 \), and \( c = 1 \). Substitute the values of \( a \), \( b \), and \( c \) into 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 of the equation are \[ x = \frac{3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{5}}{2}. \]
Conclusion: The roots of the equation are \( x = \frac{3 + \sqrt{5}}{2} \) and \( x = \frac{3 - \sqrt{5}}{2} \).