To solve the quadratic equation \( x^2 + 4x + 1 = 0 \), we can apply the quadratic formula, which is given by:

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

For the equation \( x^2 + 4x + 1 = 0 \), we have:
- \( a = 1 \)
- \( b = 4 \)
- \( c = 1 \)

### Step 1: Calculate the discriminant
The discriminant \( \Delta \) is:
\[
\Delta = b^2 - 4ac = 4^2 - 4(1)(1) = 16 - 4 = 12
\]

### Step 2: Apply the quadratic formula
Now, substitute the values of \( a \), \( b \), and \( \Delta \) into the quadratic formula:
\[
x = \frac{-4 \pm \sqrt{12}}{2(1)} = \frac{-4 \pm 2\sqrt{3}}{2}
\]

### Step 3: Simplify the expression
\[
x = \frac{-4}{2} \pm \frac{2\sqrt{3}}{2}
\]
\[
x = -2 \pm \sqrt{3}
\]

### Final answer:
The two solutions are:
\[
x_1 = -2 + \sqrt{3}, \quad x_2 = -2 - \sqrt{3}
\]

Let me know if you'd like any further details or have any questions!

Here are some related questions:
1. How do you derive the quadratic formula?
2. Can you explain how the discriminant affects the nature of the roots?
3. What are some real-life applications of quadratic equations?
4. How would the solution change if \( \Delta \) were negative?
5. How can quadratic equations be solved by completing the square?

**Tip**: Always check the discriminant (\( b^2 - 4ac \)) before solving to quickly determine the type of solutions you'll have!

Step-by-step solution of the quadratic equation x^2 + 4x + 1 using the quadratic formula. Learn how to calculate the discriminant and find the roots. Suitable for students in Grades 8-10.

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