To solve the quadratic equation [tex]\(x^2 - 6x + 2 = 0\)[/tex], we will use the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are:
[tex]\[
a = 1, \quad b = -6, \quad c = 2
\][/tex]
First, we calculate the discriminant [tex]\(D\)[/tex]:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substituting the values:
[tex]\[
D = (-6)^2 - 4 \cdot 1 \cdot 2 = 36 - 8 = 28
\][/tex]
The discriminant [tex]\(D\)[/tex] is 28.
Now we use the quadratic formula to find the roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{28}}{2a} = \frac{6 \pm \sqrt{28}}{2}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{6 \pm 2\sqrt{7}}{2} = 3 \pm \sqrt{7}
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 - 6x + 2 = 0\)[/tex] are:
[tex]\[
x_1 = 3 + \sqrt{7} \quad \text{and} \quad x_2 = 3 - \sqrt{7}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B. \; x = 3 \pm \sqrt{7}}
\][/tex]
