*x*2+18=

−6*x*

Answer

Attempt 2 out of 2

The given equation is:

\[
x^2 + 18 = -6x
\]

To solve this, let's rewrite it in the standard quadratic form \( ax^2 + bx + c = 0 \):

\[
x^2 + 6x + 18 = 0
\]

Now, solve using the **quadratic formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where:
- \( a = 1 \)
- \( b = 6 \)
- \( c = 18 \)

Substituting into the formula:
\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(18)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 - 72}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{-36}}{2}
\]
\[
x = \frac{-6 \pm 6i}{2}
\]

Simplifying:
\[
x = -3 \pm 3i
\]

The solutions are:
\[
x = -3 + 3i \quad \text{or} \quad x = -3 - 3i
\]

These are complex solutions.

Would you like more details on this process or have any questions?

Here are some related questions:
1. What are complex numbers and how do they arise?
2. Can every quadratic equation have real solutions?
3. How do you use the quadratic formula step by step?
4. What are imaginary numbers and how are they used in math?
5. Why is it important to factor quadratics before applying the quadratic formula?

**Tip:** When solving quadratics, always check if the discriminant \(b^2 - 4ac\) is negative, which leads to complex (imaginary) solutions.

Explore how to solve the quadratic equation x^2 + 6x + 18 = 0, involving complex solutions. This step-by-step guide demonstrates the use of the quadratic formula and explains the concept of complex numbers. Ideal for students in Grades 9-11.

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