To solve the quadratic equation [tex]\(3x^2 - 4x + 10 = 0\)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 10\)[/tex]
### Step-by-Step Solution:
1. Calculate the Discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Plugging in the coefficients:
[tex]\[
\text{Discriminant} = (-4)^2 - 4 \cdot 3 \cdot 10 = 16 - 120 = -104
\][/tex]
2. Determine the Nature of the Roots:
Since the discriminant is negative ([tex]\(-104\)[/tex]), the quadratic equation has two complex (imaginary) roots.
3. Compute the Square Root of the Discriminant:
[tex]\[
\sqrt{-104} = \sqrt{104} \cdot i = 2 \sqrt{26} \cdot i
\][/tex]
4. Apply the Quadratic Formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the values:
[tex]\[
x = \frac{-(-4) \pm \sqrt{-104}}{2 \cdot 3} = \frac{4 \pm 2i\sqrt{26}}{6}
\][/tex]
5. Simplify the Expression:
[tex]\[
x = \frac{4 \pm 2i\sqrt{26}}{6} = \frac{4}{6} \pm \frac{2i\sqrt{26}}{6} = \frac{2}{3} \pm \frac{i\sqrt{26}}{3}
\][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[
x = \frac{2}{3} \pm \frac{i \sqrt{26}}{3}
\][/tex]
### Matching with the Given Choices:
The correct choice is:
C) [tex]\( x = \frac{2 \pm i \sqrt{26}}{3} \)[/tex]
Thus, the answer is:
C) Choice C
