https://file.aiquickdraw.com/mathbot/file/1726117742398m4issuft.jpg

The given image presents a quadratic equation problem asking to determine the type of solution for the equation \(2x^2 + 3x + 2 = 0\).

### Step 1: Determine the discriminant
For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is given by:
\[
\Delta = b^2 - 4ac
\]
In this case, \(a = 2\), \(b = 3\), and \(c = 2\). Plugging these values into the discriminant formula:

\[
\Delta = 3^2 - 4(2)(2) = 9 - 16 = -7
\]

### Step 2: Analyze the discriminant
- If \(\Delta > 0\), the equation has two distinct real solutions.
- If \(\Delta = 0\), the equation has one real solution (a repeated or double root).
- If \(\Delta < 0\), the equation has two distinct complex (imaginary) solutions.

Since \(\Delta = -7\), which is less than 0, the equation has two distinct complex (imaginary) solutions.

### Conclusion:
The answer is \( ③ \) \( \text{異なる2つの虚数解をもつ} \) ("has two distinct complex solutions").

Would you like further details or have any questions?

Here are 5 related questions you could explore:
1. How do we solve quadratic equations with imaginary solutions?
2. What is the geometric interpretation of complex roots on the complex plane?
3. How does changing the discriminant affect the roots of a quadratic equation?
4. Can a quadratic equation have both real and complex solutions?
5. What happens if the discriminant equals zero?

**Tip:** The discriminant is key in predicting the nature of roots without fully solving the quadratic equation!

次の２次方程式の解の種類を判別せよ。
２次方程式 2x^2 + 3x + 2 = 0 は、どの種類の解を持つか。

Learn how to determine the type of roots for the quadratic equation 2x^2 + 3x + 2 = 0 using discriminant analysis. Explore key algebraic concepts and theorems relevant to solving quadratic equations, suitable for students in Grades 8-10.

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