Solve the following \( \frac{3}{x}+x+2 x=2 \)

To solve the equation \( \frac{3}{x} + x + 2x = 2 \), first combine the \(x\) terms: \[ \frac{3}{x} + 3x = 2 \] Next, multiply through by \(x\) (assuming \(x \neq 0\)) to eliminate the fraction: \[ 3 + 3x^2 = 2x \] Rearranging gives: \[ 3x^2 - 2x + 3 = 0 \] This is a quadratic equation. You can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = -2\), and \(c = 3\): Calculating the discriminant: \[ (-2)^2 - 4(3)(3) = 4 - 36 = -32 \] Since the discriminant is negative, there are no real solutions for the equation. Thus, the roots are complex: \[ x = \frac{2 \pm \sqrt{-32}}{6} = \frac{2 \pm 4i\sqrt{2}}{6} = \frac{1 \pm 2i\sqrt{2}}{3} \] So the solutions are: \[ x = \frac{1 + 2i\sqrt{2}}{3} \quad \text{and} \quad x = \frac{1 - 2i\sqrt{2}}{3} \]