Question 12

Solve for [tex]\( x \)[/tex].

[tex]\[ 3x^2 - 4x + 10 = 0 \][/tex]

A. [tex]\( x = \frac{4 \pm \sqrt{136}}{6} \)[/tex]

B. [tex]\( x = \frac{-2 \pm i \sqrt{26}}{3} \)[/tex]

C. [tex]\( x = \frac{2 \pm i \sqrt{26}}{3} \)[/tex]

D. [tex]\( x = \frac{-4 \pm \sqrt{136}}{6} \)[/tex]

A. Choice A

B. Choice B

C. Choice C

D. Choice D



Answer :

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