Answered

What is the solution to [tex]3x^2 - 2x + 4 = 0[/tex]?

A. [tex]x = \frac{-1 \pm i \sqrt{11}}{3}[/tex]
B. [tex]x = \frac{1 \pm i \sqrt{11}}{3}[/tex]
C. [tex]x = \frac{1 \pm 2i \sqrt{11}}{3}[/tex]
D. [tex]x = \frac{1 \pm \sqrt{13}}{3}[/tex]
E. [tex]x = \frac{1 \pm i \sqrt{13}}{3}[/tex]



Answer :

GinnyAnswer
To solve the quadratic equation [tex]\(3x^2 - 2x + 4 = 0\)[/tex], we will use the quadratic formula:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

Here, the coefficients are [tex]\(a = 3\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex].

1. Calculate the discriminant:

[tex]\[ \Delta = b^2 - 4ac \][/tex]

[tex]\[ \Delta = (-2)^2 - 4 \cdot 3 \cdot 4 \][/tex]

[tex]\[ \Delta = 4 - 48 \][/tex]

[tex]\[ \Delta = -44 \][/tex]

Since the discriminant [tex]\(\Delta\)[/tex] is negative, this indicates that the roots are complex.

2. Find the roots using the quadratic formula:

[tex]\[ x = \frac{-(-2) \pm \sqrt{-44}}{2 \cdot 3} \][/tex]

[tex]\[ x = \frac{2 \pm \sqrt{-44}}{6} \][/tex]

We know that [tex]\(\sqrt{-44} = \sqrt{44} \cdot i\)[/tex].

[tex]\[ x = \frac{2 \pm \sqrt{44} \cdot i}{6} \][/tex]

Simplify the square root of 44:

[tex]\[ \sqrt{44} = \sqrt{4 \cdot 11} = 2 \sqrt{11} \][/tex]

Substituting this in:

[tex]\[ x = \frac{2 \pm 2 \sqrt{11} \cdot i}{6} \][/tex]

Divide the numerator and denominator by 2:

[tex]\[ x = \frac{1 \pm \sqrt{11} \cdot i}{3} \][/tex]

Thus, the solutions to the equation [tex]\(3x^2 - 2x + 4 = 0\)[/tex] are:

[tex]\[ x = \frac{1 \pm i\sqrt{11}}{3} \][/tex]

Therefore, the correct option is:

B. [tex]\(x = \frac{1 \pm i\sqrt{11}}{3}\)[/tex]

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