https://file.aiquickdraw.com/mathbot/file/17294400382538t0ni09m.jpg

The equation provided in the image is:

\[
3x^2 - 2x + 5 = 0
\]

To find the complex solutions of this quadratic equation, we can use the **quadratic formula**:

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

For this equation, the coefficients are:
- \( a = 3 \)
- \( b = -2 \)
- \( c = 5 \)

Now, let's calculate the discriminant (\( \Delta \)) first:

\[
\Delta = b^2 - 4ac = (-2)^2 - 4(3)(5) = 4 - 60 = -56
\]

Since the discriminant is negative (\( -56 \)), the solutions will be complex. Now, applying the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{-56}}{2(3)} = \frac{2 \pm \sqrt{-56}}{6}
\]

We know that \( \sqrt{-56} = \sqrt{56}i \), so:

\[
x = \frac{2 \pm \sqrt{56}i}{6} = \frac{2}{6} \pm \frac{\sqrt{56}i}{6} = \frac{1}{3} \pm \frac{\sqrt{56}}{6}i
\]

Finally, the two complex solutions are:

\[
x = \frac{1}{3} \pm \frac{\sqrt{56}}{6}i
\]

Simplifying \( \sqrt{56} \), we get \( \sqrt{56} = 2\sqrt{14} \), so the solutions become:

\[
x = \frac{1}{3} \pm \frac{\sqrt{14}}{3}i
\]

Thus, the complex solutions are:

\[
x = \frac{1}{3} + \frac{\sqrt{14}}{3}i, \quad x = \frac{1}{3} - \frac{\sqrt{14}}{3}i
\]

Would you like more details on any part of this process? 

Here are some related questions for further exploration:
1. How is the discriminant used to determine the nature of solutions?
2. What are the conditions for a quadratic equation to have purely real solutions?
3. How can complex conjugate pairs be interpreted geometrically on the complex plane?
4. Could the quadratic formula be applied differently for special cases of \(a = 1\)?
5. How would solving the same equation graphically represent its complex solutions?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before solving quadratics to determine if the solutions will be real or complex.

Find all complex solutions of 3x^2 - 2x + 5 = 0.

Solve the quadratic equation 3x^2 - 2x + 5 = 0 by finding its complex solutions using the quadratic formula. This problem demonstrates the use of the discriminant to determine complex roots, suitable for students in Grades 9-12.

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