x2+3x+8=0

To solve the quadratic equation \( x^2 + 3x + 8 = 0 \), we can use the quadratic formula:

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

where \( a = 1 \), \( b = 3 \), and \( c = 8 \).

1. Calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 8 = 9 - 32 = -23 \]

2. Since the discriminant is negative, this means that there are no real solutions, but there are two complex solutions.

3. Now, plug the values into the quadratic formula: \[ x = \frac{{-3 \pm \sqrt{{-23}}}}{2 \cdot 1} = \frac{{-3 \pm i\sqrt{23}}}{2} \]

Thus, the solutions to the equation \( x^2 + 3x + 8 = 0 \) are: \[ x = \frac{{-3 + i\sqrt{23}}}{2} \quad \text{and} \quad x = \frac{{-3 - i\sqrt{23}}}{2} \]