To solve the quadratic equation [tex]\(3x^2 - 5x - 13 = 0\)[/tex], we will use the quadratic formula. 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 from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex].
For our equation [tex]\(3x^2 - 5x - 13 = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = -13\)[/tex]
Let's start by substituting these values into the quadratic formula.
First, calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the coefficients into the discriminant formula:
[tex]\[
\Delta = (-5)^2 - 4(3)(-13)
\][/tex]
Evaluate the terms:
[tex]\[
\Delta = 25 + 156
\][/tex]
[tex]\[
\Delta = 181
\][/tex]
Now that we have the discriminant, we can substitute back into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(b = -5\)[/tex], [tex]\(\Delta = 181\)[/tex], and [tex]\(a = 3\)[/tex]:
[tex]\[
x = \frac{-(-5) \pm \sqrt{181}}{2(3)}
\][/tex]
Simplify:
[tex]\[
x = \frac{5 \pm \sqrt{181}}{6}
\][/tex]
Therefore, the solutions to the equation [tex]\(3x^2 - 5x - 13 = 0\)[/tex] are:
[tex]\[
x = \frac{5 \pm \sqrt{181}}{6}
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{\frac{5 \pm \sqrt{181}}{6}}
\][/tex]
