To solve the quadratic equation [tex]\(x^2 + 3x + 1 = 0\)[/tex], we'll use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = 1\)[/tex]. Now, let's follow the steps carefully:
1. Calculate the Discriminant:
The discriminant [tex]\(D\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substitute the coefficients [tex]\(a = 1\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = 1\)[/tex] into the discriminant formula:
[tex]\[
D = 3^2 - 4 \cdot 1 \cdot 1
\][/tex]
[tex]\[
D = 9 - 4
\][/tex]
[tex]\[
D = 5
\][/tex]
2. Calculate the Roots:
Next, we substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant [tex]\(D\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-3 \pm \sqrt{5}}{2}
\][/tex]
This gives us the two solutions:
[tex]\[
x_1 = \frac{-3 + \sqrt{5}}{2}
\][/tex]
and
[tex]\[
x_2 = \frac{-3 - \sqrt{5}}{2}
\][/tex]
3. Evaluate the Roots:
To get the numerical values of the roots, we compute:
[tex]\[
x_1 = \frac{-3 + \sqrt{5}}{2} \approx -0.3819660112501051
\][/tex]
and
[tex]\[
x_2 = \frac{-3 - \sqrt{5}}{2} \approx -2.618033988749895
\][/tex]
Thus, the quadratic equation [tex]\(x^2 + 3x + 1 = 0\)[/tex] has the following solutions:
[tex]\[
x = \frac{-3 \pm \sqrt{5}}{2}
\][/tex]
Given the answer options, the correct answer is:
D. [tex]\(x=\frac{-3 \pm \sqrt{5}}{2}\)[/tex]
