https://file.aiquickdraw.com/mathbot/file/1726407446163cbgw51az.jpg

The equation you are solving in the image is a quadratic equation: 

\[
x^2 + 4x + 5 = 0
\]

You correctly identified the coefficients: 
- \(a = 1\)
- \(b = 4\)
- \(c = 5\)

The next step is to apply the **quadratic formula**:

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

Substitute the values \(a = 1\), \(b = 4\), and \(c = 5\) into the formula:

\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(5)}}{2(1)}
\]

This simplifies to:

\[
x = \frac{-4 \pm \sqrt{16 - 20}}{2}
\]
\[
x = \frac{-4 \pm \sqrt{-4}}{2}
\]

Now, you encounter a square root of a negative number, \(\sqrt{-4}\), which implies that the solutions are **complex** numbers. In complex numbers, \(\sqrt{-1} = i\), where \(i\) is the imaginary unit. Therefore:

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

Now substitute this back into the formula:

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

Simplify:

\[
x = -2 \pm i
\]

Thus, the two complex solutions are:

\[
x = -2 + i \quad \text{or} \quad x = -2 - i
\]

These are the final solutions to the quadratic equation.

---

Would you like any further explanation or details on this?

Here are five related questions that expand upon this topic:
1. What are the conditions for a quadratic equation to have real vs complex solutions?
2. How do you graph quadratic equations with complex solutions?
3. Can the discriminant (\(b^2 - 4ac\)) predict the nature of the roots?
4. How do imaginary numbers like \(i\) behave in calculations involving powers?
5. What are the applications of complex numbers in physics or engineering?

**Tip:** The discriminant (\(b^2 - 4ac\)) is key to knowing whether your quadratic has real or complex solutions.

How to solve the quadratic equation x^2 + 4x + 5 = 0?

This page provides a detailed explanation of how to solve the quadratic equation x^2 + 4x + 5 = 0 using the quadratic formula. It demonstrates that the equation has complex solutions, explains the role of the discriminant, and introduces complex numbers (i). Suitable for high school students.

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