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!