To solve the given quadratic equation [tex]\(2x^2 - 3x + 2 = 0\)[/tex], we can follow a structured approach:
1. Identify the coefficients: Here, we have [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 2\)[/tex].
2. Calculate the discriminant: The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values, we get:
[tex]\[
\Delta = (-3)^2 - 4 \cdot 2 \cdot 2 = 9 - 16 = -7
\][/tex]
Since the discriminant is negative ([tex]\(\Delta = -7\)[/tex]), we know that the roots of the quadratic equation are complex numbers.
3. Calculate the real and imaginary parts: For complex roots, they are given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Here, [tex]\(\Delta = -7\)[/tex]. Breaking it down, we get:
[tex]\[
x = \frac{-(-3) \pm \sqrt{-7}}{2 \cdot 2} = \frac{3 \pm \sqrt{-7}}{4}
\][/tex]
Recall that [tex]\(\sqrt{-7} = i\sqrt{7}\)[/tex], thus:
[tex]\[
x = \frac{3 \pm i\sqrt{7}}{4}
\][/tex]
4. Write down the roots: Therefore, the roots are:
[tex]\[
x_1 = \frac{3 + i\sqrt{7}}{4}
\][/tex]
[tex]\[
x_2 = \frac{3 - i\sqrt{7}}{4}
\][/tex]
These roots can thus be expressed in the solution set:
[tex]\[
\left\{\frac{3 + i \sqrt{7}}{4}, \frac{3 - i \sqrt{7}}{4}\right\}
\][/tex]
Comparing this with the given options, we see that the correct answer is:
[tex]\[
\text{B. }\left\{\frac{3 + i \sqrt{7}}{4}, \frac{3 - i \sqrt{7}}{4}\right\}
\][/tex]
