Solve the quadratic equation $x^2 + 4x + 1 = 0$ by completing the square.

1. Problem Description

Solve the quadratic equation $x^2 + 4x + 1 = 0$ by completing the square.

2. Solution Steps

Step 1: Move the constant term to the right side of the equation.

$x^2 + 4x = -1$

Step 2: Complete the square on the left side. Take half of the coefficient of the $x$ term, square it, and add it to both sides of the equation. The coefficient of the $x$ term is

4. Half of 4 is 2, and $2^2 = 4$. Add 4 to both sides.

$x^2 + 4x + 4 = -1 + 4$

Step 3: Factor the left side as a perfect square and simplify the right side.

$(x + 2)^2 = 3$

Step 4: Take the square root of both sides.

$x + 2 = \pm\sqrt{3}$

Step 5: Solve for $x$.

$x = -2 \pm\sqrt{3}$

3. Final Answer

$x = -2 + \sqrt{3}$ or $x = -2 - \sqrt{3}$