depending on the value of k, the equation x^2 + kx + 2024 = 0, for how many positive interger values of k does the equation have 2 integer solutions

To determine the number of positive integer values of \( k \) for which the quadratic equation \( x^2 + kx + 2024 = 0 \) has two integer solutions, we follow these steps:

1. **Identify the General Form of the Quadratic Equation**:  
   The given equation is \( x^2 + kx + 2024 = 0 \).

2. **Condition for Integer Solutions**:  
   For the quadratic equation \( ax^2 + bx + c = 0 \) to have integer solutions, the discriminant \( D \) must be a perfect square. The discriminant \( D \) for our equation is given by:
   \[ D = b^2 - 4ac \]
   Here, \( a = 1 \), \( b = k \), and \( c = 2024 \). Thus:
   \[ D = k^2 - 4 \times 1 \times 2024 = k^2 - 8096 \]

3. **Perfect Square Condition**:  
   \( k^2 - 8096 \) must be a perfect square. Let \( D = m^2 \) where \( m \) is an integer. Then:
   \[ k^2 - 8096 = m^2 \]
   \[ k^2 - m^2 = 8096 \]
   \[ (k - m)(k + m) = 8096 \]

4. **Factorizing 8096**:  
   To find all possible pairs \( (k - m) \) and \( (k + m) \), we first factorize 8096:
   \[ 8096 = 2^4 \times 1012 \]
   \[ 1012 = 2^2 \times 253 \]
   \[ 253 = 11 \times 23 \]
   Therefore, the prime factorization of 8096 is:
   \[ 8096 = 2^6 \times 11 \times 23 \]

   Using the factorization, we can find all pairs \( (a, b) \) such that \( a \times b = 8096 \). We then set \( k - m = a \) and \( k + m = b \) to determine \( k \) and \( m \).

5. **Calculation of k and m**:  
   We calculate \( k \) and \( m \) as follows:
   \[ k = \frac{(k + m) + (k - m)}{2} = \frac{a + b}{2} \]
   \[ m = \frac{(k + m) - (k - m)}{2} = \frac{b - a}{2} \]

   We will only consider the pairs \( (a, b) \) such that both \( \frac{a + b}{2} \) and \( \frac{b - a}{2} \) are integers.

6. **Counting the Positive Integer Values of k**:  
   We will find all such pairs \( (a, b) \) and count the number of valid \( k \) values.

Let's go ahead and find the factors and then determine the corresponding values for \( k \).There are **8 positive integer values of \( k \)** for which the equation \( x^2 + kx + 2024 = 0 \) has two integer solutions.

### Additional Questions and Tip

1. What are the actual values of \( k \) for which the equation has integer solutions?
2. Can the equation have two identical integer solutions? Why or why not?
3. What would happen if the coefficient of \( x \) was negative?
4. How would the nature of the solutions change if the constant term was different?
5. What is the significance of the discriminant in determining the nature of the roots?
6. How can you determine the roots of the quadratic equation once you know \( k \)?
7. What are the conditions under which a quadratic equation has no real roots?
8. How does the symmetry of the parabola affect the solutions of the quadratic equation?

**Tip**: When solving for integer roots of a quadratic equation, always check if the discriminant is a perfect square, as this ensures the roots are real and rational.

Discover how many positive integer values of k result in two integer solutions for the quadratic equation x^2 + kx + 2024 = 0. Explore the discriminant and factorization methods to find solutions. Suitable for advanced high school students (Grades 10-12).

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