2х^2+6х+6=0

To solve the quadratic equation \(2x^2 + 6x + 6 = 0\), we can use the quadratic formula:

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

Here, \(a = 2\), \(b = 6\), and \(c = 6\).

First, we calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 - 4ac = 6^2 - 4 \cdot 2 \cdot 6 = 36 - 48 = -12 \]

Since the discriminant is negative, the equation has no real solutions, but there are two complex solutions.

Now, we'll substitute back into the quadratic formula:

\[ x = \frac{-6 \pm \sqrt{-12}}{2 \cdot 2} \]

Simplifying further:

\[ \sqrt{-12} = \sqrt{12}i = 2\sqrt{3}i \]

So we substitute that back into the equation:

\[ x = \frac{-6 \pm 2\sqrt{3}i}{4} \]

Now, simplify the fraction:

\[ x = \frac{-3 \pm \sqrt{3}i}{2} \]

Thus, the solutions to the equation \(2x^2 + 6x + 6 = 0\) are:

\[ x = \frac{-3 + \sqrt{3}i}{2} \quad \text{and} \quad x = \frac{-3 - \sqrt{3}i}{2} \]