To solve the quadratic equation \(x^2 = -5x - 3\), we can rewrite it in standard form:
\[ x^2 + 5x + 3 = 0 \]
Now we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 5\), and \(c = 3\).
First, we need to calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13 \]
Now we can substitute back into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{13}}{2 \cdot 1} = \frac{-5 \pm \sqrt{13}}{2} \]
This gives us the two solutions:
- \(x = \frac{-5 - \sqrt{13}}{2}\)
- \(x = \frac{-5 + \sqrt{13}}{2}\)
Among the options provided, the correct ones are:
\[ \frac{-5 - \sqrt{13}}{2}, \frac{-5 + \sqrt{13}}{2} \]
So, the solutions of the quadratic equation \(x^2 = -5x - 3\) are:
\[ \text{StartFraction negative 5 minus StartRoot 13 EndRoot Over 2 EndFraction, StartFraction negative 5 plus StartRoot 13 EndRoot Over 2 EndFraction} \]