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You need to find all complex solutions of the quadratic equation \( 2x^2 - 3x + 6 = 0 \).

### Step-by-step solution:
We will use the quadratic formula to solve this:

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

For the equation \( 2x^2 - 3x + 6 = 0 \), we identify:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 6 \)

#### Step 1: Calculate the discriminant
\[
\Delta = b^2 - 4ac = (-3)^2 - 4(2)(6) = 9 - 48 = -39
\]
The discriminant is negative, so the solutions will be complex.

#### Step 2: Use the quadratic formula
\[
x = \frac{-(-3) \pm \sqrt{-39}}{2(2)} = \frac{3 \pm \sqrt{-39}}{4}
\]
\[
x = \frac{3 \pm \sqrt{39}i}{4}
\]

#### Step 3: Express the two complex solutions
\[
x_1 = \frac{3 + \sqrt{39}i}{4}, \quad x_2 = \frac{3 - \sqrt{39}i}{4}
\]

### Final answer:
\[
x = \frac{3 + \sqrt{39}i}{4}, \frac{3 - \sqrt{39}i}{4}
\]

Let me know if you'd like more details or have any questions!

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Here are 5 related questions to explore:
1. How do we interpret the discriminant in relation to complex solutions?
2. What does the presence of an imaginary unit \(i\) indicate in the solution?
3. How would the solution change if the discriminant was positive?
4. What is the geometric representation of these complex solutions on the complex plane?
5. Can you apply this method to solve any quadratic equation?

**Tip:** When using the quadratic formula, always check the discriminant first to know whether the solutions are real or complex.

Find all complex solutions of 2x^2 - 3x + 6 = 0.

Explore how to find the complex solutions of the quadratic equation 2x^2 - 3x + 6 = 0 using the quadratic formula. This solution demonstrates the use of imaginary numbers and is ideal for students in high school mathematics.

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