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"Resuelve la siguiente ecuación cuadrática: \( 2x^2 - 5x + 3 = 0 \) utilizando la fórmula general. Escribe el discriminante, las soluciones y simplifica si es necesario."

Let's solve this step by step using the quadratic formula.

### 1. **Quadratic Formula**:
The general quadratic equation is given by:
\[
ax^2 + bx + c = 0
\]
The solutions are given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### 2. **Identify coefficients**:
For the equation \( 2x^2 - 5x + 3 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -5 \)
- \( c = 3 \)

### 3. **Calculate the Discriminant**:
The discriminant \( \Delta \) is given by:
\[
\Delta = b^2 - 4ac
\]
Substituting the values:
\[
\Delta = (-5)^2 - 4(2)(3) = 25 - 24 = 1
\]

### 4. **Find the Solutions**:
Since the discriminant \( \Delta = 1 \) is positive, we have two real and distinct solutions. Using the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{1}}{2(2)} = \frac{5 \pm 1}{4}
\]
This gives us two solutions:
\[
x_1 = \frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2}
\]
\[
x_2 = \frac{5 - 1}{4} = \frac{4}{4} = 1
\]

### **Final Answer**:
The solutions to the quadratic equation \( 2x^2 - 5x + 3 = 0 \) are:
\[
x_1 = \frac{3}{2}, \quad x_2 = 1
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How can you determine the nature of the roots using the discriminant?
2. What happens if the discriminant is negative?
3. Can the quadratic formula be used for any degree polynomial?
4. What are the other methods to solve quadratic equations?
5. How does changing the coefficients affect the solutions?

### Tip:
Always check the discriminant before solving the quadratic equation to understand the nature of the roots quickly.

Resuelve la siguiente ecuación cuadrática: 2x^2 - 5x + 3 = 0 utilizando la fórmula general. Escribe el discriminante, las soluciones y simplifica si es necesario.

Learn how to solve the quadratic equation 2x^2 - 5x + 3 = 0 using the general formula. This problem explains key algebraic concepts, including the quadratic formula and the discriminant. Suitable for students in Grades 8-10.

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