Initial Knowledge Check

Find all complex solutions of $2 x^{2}-3 x+6=0$.
(If there is more than one solution, separate them with commas.)
\[
x=[\pi]
\]

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 ：\(\boxed{x = 0.75 - 1.5612494995995996j, x = 0.75 + 1.5612494995995996j}\)