Solve by Completing the Square x^2+4x+1=0

$x^{2} + 4 x + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(2)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 4 x + {(2)}^{2} = - 1 + {(2)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $2$ to the power of $2$ .

$x^{2} + 4 x + 4 = - 1 + {(2)}^{2}$

Simplify the right side.

Simplify $- 1 + {(2)}^{2}$ .

Raise $2$ to the power of $2$ .

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

Add $- 1$ and $4$ .

$x^{2} + 4 x + 4 = 3$

Step 5

Factor the perfect trinomial square into ${(x + 2)}^{2}$ .

${(x + 2)}^{2} = 3$

Step 6

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Subtract $2$ from both sides of the equation.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.26794919 \dots, - 3.73205080 \dots$