X^2+7x+13=0

To solve the quadratic equation \( x^2 + 7x + 13 = 0 \), we can use the quadratic formula:

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

In this equation, \( a = 1 \), \( b = 7 \), and \( c = 13 \).

First, we calculate the discriminant:

\[ b^2 - 4ac = 7^2 - 4 \cdot 1 \cdot 13 = 49 - 52 = -3 \]

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

Now we substitute into the quadratic formula:

\[ x = \frac{-7 \pm \sqrt{-3}}{2 \cdot 1} \] \[ x = \frac{-7 \pm i\sqrt{3}}{2} \]

Thus, the solutions are:

\[ x = \frac{-7 + i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{-7 - i\sqrt{3}}{2} \]

These are the complex solutions to the equation \( x^2 + 7x + 13 = 0 \).