To find the value of \( x \) in the equation \( x^2 + 8x + 28 = 0 \), we can use the quadratic formula:

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

For the equation \( x^2 + 8x + 28 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 8 \)
- \( c = 28 \)

Now, substitute these values into the quadratic formula:

\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(28)}}{2(1)}
\]

Simplify the expression:

\[
x = \frac{-8 \pm \sqrt{64 - 112}}{2}
\]

\[
x = \frac{-8 \pm \sqrt{-48}}{2}
\]

Since the discriminant (\( \sqrt{-48} \)) is negative, the solutions will be complex numbers. We can express \( \sqrt{-48} \) as \( \sqrt{48} \times i \), where \( i \) is the imaginary unit:

\[
x = \frac{-8 \pm \sqrt{48} \times i}{2}
\]

Simplify \( \sqrt{48} \) and then the whole expression:

\[
\sqrt{48} = 4\sqrt{3}
\]

\[
x = \frac{-8 \pm 4\sqrt{3} \times i}{2}
\]

Finally, divide by 2:

\[
x = -4 \pm 2\sqrt{3}i
\]

So the solutions are:

\[
x = -4 + 2\sqrt{3}i \quad \text{and} \quad x = -4 - 2\sqrt{3}i
\]

Would you like further details on any part of this solution?

Here are some related questions:

1. How can you determine the nature of the roots using the discriminant?
2. What is the geometric interpretation of complex roots on the complex plane?
3. How do you add or subtract complex numbers?
4. Can you solve this quadratic equation by completing the square instead?
5. What are the properties of the imaginary unit \( i \)?

**Tip:** When solving quadratic equations, the discriminant \( b^2 - 4ac \) is crucial in determining whether the roots are real or complex.

Learn how to solve the quadratic equation x^2 + 8x + 28 = 0 with complex solutions using the quadratic formula. Understand the nature of roots based on the discriminant. Suitable for high school students studying algebra and complex numbers.

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