To find the roots of the quadratic equation [tex]\( x^2 - 5x + 2 = 0 \)[/tex], we can use the quadratic formula, which states:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the given equation [tex]\( x^2 - 5x + 2 = 0 \)[/tex], we identify the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -5 \)[/tex]
- [tex]\( c = 2 \)[/tex]
Step 1: Compute the discriminant [tex]\(\Delta\)[/tex].
[tex]\[
\Delta = b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 2 = 25 - 8 = 17
\][/tex]
Step 2: Calculate the roots using the quadratic formula.
[tex]\[
x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\( b = -5 \)[/tex] and [tex]\( \Delta = 17 \)[/tex]:
[tex]\[
x = \frac{5 \pm \sqrt{17}}{2}
\][/tex]
Thus, the roots are:
[tex]\[
x_1 = \frac{5 + \sqrt{17}}{2}, \quad x_2 = \frac{5 - \sqrt{17}}{2}
\][/tex]
Step 3: Identify the correct options given in the problem.
The equations corresponding to the given roots and answers are:
- Option A: [tex]\( x = \frac{5 + \sqrt{17}}{2} \)[/tex]
- Option B: [tex]\( x = \frac{5 - \sqrt{17}}{2} \)[/tex]
- Option C: [tex]\( x = \frac{5 + \sqrt{33}}{2} \)[/tex]
- Option D: [tex]\( x = \frac{5 - \sqrt{33}}{2} \)[/tex]
From the calculations:
The correct roots are:
[tex]\[
x_1 = \frac{5 + \sqrt{17}}{2}, \quad x_2 = \frac{5 - \sqrt{17}}{2}
\][/tex]
Matching these with the given options, we identify the correct options as:
- Option A: [tex]\( x = \frac{5 + \sqrt{17}}{2} \)[/tex]
- Option B: [tex]\( x = \frac{5 - \sqrt{17}}{2} \)[/tex]
Therefore, the correct selections for the roots are:
[tex]\[
\boxed{A \text{ and } B}
\][/tex]
