To solve the quadratic equation [tex]\( x^2 + 4x + 1 = 0 \)[/tex], we will use the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 1 \)[/tex]
First, let's calculate the discriminant ([tex]\( \Delta \)[/tex]), which is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 4^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12
\][/tex]
The discriminant is 12.
Next, we use the quadratic formula to find the roots. We need to calculate each root separately using the positive and negative forms of the [tex]\(\pm\)[/tex] sign in the formula.
The first root ([tex]\( x_1 \)[/tex]) is calculated as:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-4 + \sqrt{12}}{2 \cdot 1} = \frac{-4 + \sqrt{12}}{2}
\][/tex]
Solving this, we get:
[tex]\[
x_1 = \frac{-4 + 3.4641016151377544}{2} = \frac{-0.5358983848622456}{2} = -0.2679491924311228
\][/tex]
Rounding [tex]\( x_1 \)[/tex] to two decimal places:
[tex]\[
x_1 \approx -0.27
\][/tex]
The second root ([tex]\( x_2 \)[/tex]) is calculated as:
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-4 - \sqrt{12}}{2 \cdot 1} = \frac{-4 - \sqrt{12}}{2}
\][/tex]
Solving this, we get:
[tex]\[
x_2 = \frac{-4 - 3.4641016151377544}{2} = \frac{-7.4641016151377544}{2} = -3.732050807568877
\][/tex]
Rounding [tex]\( x_2 \)[/tex] to two decimal places:
[tex]\[
x_2 \approx -3.73
\][/tex]
Therefore, the roots of the quadratic equation [tex]\( x^2 + 4x + 1 = 0 \)[/tex], rounded to two decimal places, are:
[tex]\[
\boxed{-0.27 \text{ and } -3.73}
\][/tex]
