To solve the quadratic equation [tex]\( 3x^2 + 10 = 4x \)[/tex], we will follow these steps:
1. Rearrange the equation into standard form [tex]\( ax^2 + bx + c = 0 \)[/tex].
2. Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] to find the roots.
### Step 1: Rearrange the Equation
First, we need to write the equation in the standard form:
[tex]\[
3x^2 + 10 = 4x
\][/tex]
Subtract [tex]\( 4x \)[/tex] from both sides:
[tex]\[
3x^2 - 4x + 10 = 0
\][/tex]
### Step 2: Apply the Quadratic Formula
For the equation [tex]\( 3x^2 - 4x + 10 = 0 \)[/tex], we identify the coefficients:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 10\)[/tex]
Plug these values into the quadratic formula:
[tex]\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{4 \pm \sqrt{16 - 120}}{6}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{-104}}{6}
\][/tex]
Since the discriminant ([tex]\(-104\)[/tex]) is negative, the equation has complex roots. We can write [tex]\(\sqrt{-104}\)[/tex] as [tex]\( \sqrt{104} \cdot i \)[/tex] where [tex]\(i\)[/tex] is the imaginary unit:
[tex]\[
\sqrt{-104} = \sqrt{4 \cdot 26} \cdot i = 2 \sqrt{26} \cdot i
\][/tex]
Substitute back into the formula:
[tex]\[
x = \frac{4 \pm 2 \sqrt{26} \cdot i}{6}
\][/tex]
### Simplify the expression
We can simplify each term in the numerator by dividing by 2:
[tex]\[
x = \frac{2 \pm \sqrt{26} \cdot i}{3}
\][/tex]
Thus, the roots of the quadratic equation [tex]\( 3x^2 - 4x + 10 = 0 \)[/tex] are:
[tex]\[
x = \frac{2 \pm \sqrt{26} \cdot i}{3}
\][/tex]
### Conclusion
Therefore, the correct answer is:
B. [tex]\( x = \frac{2 \pm i \sqrt{26}}{3} \)[/tex]
