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The equation provided is:

\[
x^2 = -8x - 18
\]

We need to solve it using the quadratic formula. First, we rewrite the equation in standard quadratic form:

\[
x^2 + 8x + 18 = 0
\]

The quadratic formula is given by:

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

Here, \(a = 1\), \(b = 8\), and \(c = 18\).

Now, let's calculate the discriminant \(\Delta = b^2 - 4ac\):

\[
\Delta = 8^2 - 4(1)(18) = 64 - 72 = -8
\]

Since the discriminant is negative, the solutions will be complex numbers. We proceed as follows:

\[
x = \frac{-8 \pm \sqrt{-8}}{2(1)} = \frac{-8 \pm \sqrt{-8}}{2}
\]

We know that \(\sqrt{-8} = 2i\sqrt{2}\), where \(i\) is the imaginary unit. So the solutions become:

\[
x = \frac{-8 \pm 2i\sqrt{2}}{2}
\]

Simplifying:

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

Thus, the two solutions are:

\[
x = -4 + i\sqrt{2} \quad \text{and} \quad x = -4 - i\sqrt{2}
\]

These are the complex solutions for the given quadratic equation.

Would you like more details or have any other questions?

Here are some related questions:
1. What is the significance of the discriminant in determining the nature of the roots?
2. How do you handle complex numbers when solving quadratic equations?
3. Can quadratic equations always be solved using the quadratic formula?
4. How would the solution differ if the discriminant were positive or zero?
5. What are some real-world applications of solving quadratic equations?

**Tip**: When the discriminant is negative, the solutions will always involve complex numbers with imaginary parts.

Solve by using the quadratic formula. x^2 = -8x - 18

Learn how to solve the quadratic equation x² = -8x - 18 by using the quadratic formula. This problem involves calculating the discriminant and working with complex solutions. Suitable for students in Grades 9-12.

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