To solve the quadratic equation [tex]\(3x^2 - 8x + 5 = 0\)[/tex], we can follow these steps:
1. Identify the coefficients a, b, and c from the equation:
The given equation is of the form [tex]\(ax^2 + bx + c = 0\)[/tex].
Here, [tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 5\)[/tex].
2. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Plug in the coefficients into the quadratic formula:
[tex]\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}
\][/tex]
Simplifying the expression inside the square root:
[tex]\[
x = \frac{8 \pm \sqrt{64 - 60}}{6}
\][/tex]
4. Calculate the discriminant:
The discriminant is [tex]\(64 - 60 = 4\)[/tex].
5. Evaluate the square root of the discriminant:
[tex]\[
\sqrt{4} = 2
\][/tex]
6. Substitute back into the quadratic formula:
[tex]\[
x = \frac{8 \pm 2}{6}
\][/tex]
7. Solve for the two possible values of [tex]\(x\)[/tex]:
- For the positive case:
[tex]\[
x = \frac{8 + 2}{6} = \frac{10}{6} = \frac{5}{3}
\][/tex]
- For the negative case:
[tex]\[
x = \frac{8 - 2}{6} = \frac{6}{6} = 1
\][/tex]
Thus, the solutions to the equation [tex]\(3x^2 - 8x + 5 = 0\)[/tex] are [tex]\(x = 1\)[/tex] and [tex]\(x = \frac{5}{3}\)[/tex].
So the correct answer from the given choices is:
[tex]\[
x = 1, \frac{5}{3}
\][/tex]
