To solve the quadratic equation [tex]\(x^2 - 5x + 3 = 0\)[/tex] using the quadratic formula, we follow these steps:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients in the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For the equation [tex]\(x^2 - 5x + 3 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = 3\)[/tex]
First, we calculate the discriminant [tex]\(D\)[/tex]:
[tex]\[
D = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
D = (-5)^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13
\][/tex]
Next, we use the quadratic formula to find the roots:
[tex]\[
x = \frac{-b \pm \sqrt{D}}{2a}
\][/tex]
Substituting [tex]\(b = -5\)[/tex], [tex]\(D = 13\)[/tex], and [tex]\(a = 1\)[/tex]:
[tex]\[
x = \frac{-(-5) \pm \sqrt{13}}{2 \cdot 1} = \frac{5 \pm \sqrt{13}}{2}
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(x^2 - 5x + 3 = 0\)[/tex] are:
[tex]\[
x = \frac{5 + \sqrt{13}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{13}}{2}
\][/tex]
Thus, the correct answer is:
D. [tex]\(x = \frac{5 \pm \sqrt{13}}{2}\)[/tex]
