2x²+x+2=0

To solve the quadratic equation \( 2x^2 + x + 2 = 0 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 2 \), \( b = 1 \), and \( c = 2 \).

First, we calculate the discriminant:

\[ b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot 2 = 1 - 16 = -15 \]

Since the discriminant is negative (\(-15\)), this means that the equation has no real solutions, but it has two complex solutions.

Now we can substitute the values into the quadratic formula:

\[ x = \frac{-1 \pm \sqrt{-15}}{2 \cdot 2} \]

\[ x = \frac{-1 \pm i\sqrt{15}}{4} \]

Thus, the solutions are:

\[ x = \frac{-1 + i\sqrt{15}}{4} \quad \text{and} \quad x = \frac{-1 - i\sqrt{15}}{4} \]

These are the two complex solutions to the equation \( 2x^2 + x + 2 = 0 \).