Step 1 :We are given the quadratic equation \(2x^2 - 3x + 6 = 0\).
Step 2 :The general form of a quadratic equation is \(ax^2 + bx + c = 0\).
Step 3 :The solutions of a quadratic equation can be found using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Step 4 :Here, \(a = 2\), \(b = -3\), and \(c = 6\). We can substitute these values into the quadratic formula to find the solutions.
Step 5 :Calculating the discriminant \(D = b^2 - 4ac = (-3)^2 - 4*2*6 = -39\).
Step 6 :Since the discriminant is negative, the solutions to the equation are complex numbers.
Step 7 :Using the quadratic formula, we find the solutions to be \(x = 0.75 - 1.5612494995995996j\) and \(x = 0.75 + 1.5612494995995996j\).
Step 8 :\(\boxed{x = 0.75 - 1.5612494995995996j, x = 0.75 + 1.5612494995995996j}\)