https://file.aiquickdraw.com/mathbot/file/1726117708045uq5ad0h1.jpg

The quadratic equation provided in the image is:

\[
x^2 - 2\sqrt{5}x + 5 = 0
\]

We are tasked with determining the type of solutions this equation has.

### Step 1: Use the discriminant formula
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, \(a = 1\), \(b = -2\sqrt{5}\), and \(c = 5\). Let's calculate the discriminant:

\[
\Delta = (-2\sqrt{5})^2 - 4(1)(5)
\]
\[
\Delta = 4 \times 5 - 4 \times 5 = 20 - 20 = 0
\]

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

Since \( \Delta = 0 \), the quadratic equation has **one real solution** (repeated root).

### Conclusion:
The correct answer is:

\[
② \text{重解をもつ} \quad \text{(The equation has a repeated root)}
\]

---

Would you like further details or clarifications? 

Here are 5 related questions for further exploration:
1. What are the roots of the quadratic equation when the discriminant is zero?
2. How do complex solutions arise in quadratic equations?
3. Can a quadratic equation have only one real root without being a perfect square trinomial?
4. What is the geometrical interpretation of a repeated root for a parabola?
5. How does changing the coefficients of a quadratic equation affect its discriminant?

**Tip**: Always check the discriminant first when determining the number and type of solutions for a quadratic equation!

次の2次方程式の解の種類を判別せよ。
2次方程式 x^2 - 2√5 x + 5 = 0, ウ。
① 異なる2つの実数解をもつ
② 重解をもつ
③ 異なる2つの虚数解をもつ

Learn how to solve the quadratic equation x^2 - 2√5 x + 5 = 0 by using the discriminant formula. This problem teaches key algebraic concepts like the nature of roots based on the discriminant value. Suitable for high school students in grades 10-12.

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