Question

Find the solution(s) for -x^2-3x-6=0

Answer 1

To solve the quadratic equation \(-x^2 - 3x - 6 = 0\), we can first rearrange it in standard form:

\[
x^2 + 3x + 6 = 0
\]

Now we'll use the quadratic formula, which is given by:

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

In our equation, \(a = 1\), \(b = 3\), and \(c = 6\). Plugging these values into the formula:

1. Calculate the discriminant:

\[
b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 6 = 9 - 24 = -15
\]

Since the discriminant is negative, this indicates that there are no real solutions. However, there will be two complex solutions.

2. Now, we compute the roots using the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{-15}}{2 \cdot 1}
\]

This simplifies to:

\[
x = \frac{-3 \pm i\sqrt{15}}{2}
\]

So the solutions are:

\[
x = \frac{-3 + i\sqrt{15}}{2} \quad \text{and} \quad x = \frac{-3 - i\sqrt{15}}{2}
\]

Thus, the final solutions for the equation \(-x^2 - 3x - 6 = 0\) are:

\[
x = \frac{-3 + i\sqrt{15}}{2}, \quad x = \frac{-3 - i\sqrt{15}}{2}
\]