Sure, let's solve the quadratic equation [tex]\(x^2 + 6x + 1 = 0\)[/tex] step-by-step.
1. Identifying the coefficients:
The quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
a = 1, \quad b = 6, \quad c = 1
\][/tex]
2. Calculating the discriminant (D):
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
D = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
D = 6^2 - 4 \cdot 1 \cdot 1 = 36 - 4 = 32
\][/tex]
3. Finding the roots using the quadratic formula:
The roots of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] are given by:
[tex]\[
x = \frac{-b \pm \sqrt{D}}{2a}
\][/tex]
Plugging in the values [tex]\(a = 1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(D = 32\)[/tex]:
- For the first root ([tex]\( + \)[/tex] sign in the quadratic formula):
[tex]\[
x_1 = \frac{-6 + \sqrt{32}}{2 \cdot 1} = \frac{-6 + \sqrt{32}}{2} = \frac{-6 + 4\sqrt{2}}{2}
\][/tex]
This simplifies to:
[tex]\[
x_1 = -3 + 2\sqrt{2} \approx -0.1715728752538097
\][/tex]
- For the second root ([tex]\( - \)[/tex] sign in the quadratic formula):
[tex]\[
x_2 = \frac{-6 - \sqrt{32}}{2 \cdot 1} = \frac{-6 - \sqrt{32}}{2} = \frac{-6 - 4\sqrt{2}}{2}
\][/tex]
This simplifies to:
[tex]\[
x_2 = -3 - 2\sqrt{2} \approx -5.82842712474619
\][/tex]
4. Summarizing the results:
[tex]\[
\text{Discriminant} (D) = 32
\][/tex]
[tex]\[
\text{First Root} = -0.1715728752538097
\][/tex]
[tex]\[
\text{Second Root} = -5.82842712474619
\][/tex]
Therefore, the roots of the equation [tex]\(x^2 + 6x + 1 = 0\)[/tex] are approximately [tex]\(-0.1715728752538097\)[/tex] and [tex]\(-5.82842712474619\)[/tex], and the discriminant [tex]\(D\)[/tex] is 32.
