Can you provide more examples of finding values that satisfy the discriminant condition?

Sure, let's look at a few more examples where we find values that satisfy the discriminant condition to ensure rational roots for quadratic equations.

### Example 1
**Quadratic Equation:** \( x^2 + kx + 1 = 0 \)

For rational roots, the discriminant \( \Delta \) must be a perfect square. Here, \( a = 1 \), \( b = k \), and \( c = 1 \):

\[
\Delta = k^2 - 4ac = k^2 - 4 \cdot 1 \cdot 1 = k^2 - 4
\]

Set \( k^2 - 4 = m^2 \):

\[
k^2 - 4 = m^2 \implies k^2 - m^2 = 4 \implies (k - m)(k + m) = 4
\]

The pairs \((k - m, k + m)\) that multiply to 4 are:

\[
(1, 4), (2, 2), (-1, -4), (-2, -2)
\]

1. For \( (k - m, k + m) = (1, 4) \):
   \[
   k - m = 1 \quad \text{and} \quad k + m = 4
   \]
   Adding these equations:
   \[
   2k = 5 \implies k = \frac{5}{2} \quad (\text{not an integer})
   \]

2. For \( (k - m, k + m) = (2, 2) \):
   \[
   k - m = 2 \quad \text{and} \quad k + m = 2
   \]
   Adding these equations:
   \[
   2k = 4 \implies k = 2
   \]

Thus, \( k = 2 \).

### Example 2
**Quadratic Equation:** \( 3x^2 + kx + 2 = 0 \)

For rational roots, the discriminant \( \Delta \) must be a perfect square. Here, \( a = 3 \), \( b = k \), and \( c = 2 \):

\[
\Delta = k^2 - 4ac = k^2 - 4 \cdot 3 \cdot 2 = k^2 - 24
\]

Set \( k^2 - 24 = m^2 \):

\[
k^2 - 24 = m^2 \implies k^2 - m^2 = 24 \implies (k - m)(k + m) = 24
\]

The pairs \((k - m, k + m)\) that multiply to 24 are:

\[
(1, 24), (2, 12), (3, 8), (4, 6), (-1, -24), (-2, -12), (-3, -8), (-4, -6)
\]

1. For \( (k - m, k + m) = (1, 24) \):
   \[
   k - m = 1 \quad \text{and} \quad k + m = 24
   \]
   Adding these equations:
   \[
   2k = 25 \implies k = \frac{25}{2} \quad (\text{not an integer})
   \]

2. For \( (k - m, k + m) = (2, 12) \):
   \[
   k - m = 2 \quad \text{and} \quad k + m = 12
   \]
   Adding these equations:
   \[
   2k = 14 \implies k = 7
   \]

3. For \( (k - m, k + m) = (3, 8) \):
   \[
   k - m = 3 \quad \text{and} \quad k + m = 8
   \]
   Adding these equations:
   \[
   2k = 11 \implies k = \frac{11}{2} \quad (\text{not an integer})
   \]

4. For \( (k - m, k + m) = (4, 6) \):
   \[
   k - m = 4 \quad \text{and} \quad k + m = 6
   \]
   Adding these equations:
   \[
   2k = 10 \implies k = 5
   \]

Thus, \( k = 7 \) and \( k = 5 \) are the values for which the discriminant condition is satisfied, resulting in rational roots. The least positive \( k \) is 5.

Do you have any questions or need further details?

Here are some questions you might ask:
1. Can you explain how the discriminant determines the nature of roots?
2. What are some other methods to find rational roots of quadratic equations?
3. How do you solve quadratic equations using the quadratic formula?
4. Can you show an example of a quadratic equation with irrational roots?
5. What are the real-world applications of quadratic equations?

**Tip:** When working with quadratic equations, always consider the discriminant first to understand the nature of the roots before solving the equation.

Explore more examples of finding values that satisfy the discriminant condition to ensure rational roots for quadratic equations. Detailed examples include step-by-step solutions and explanations. Suitable for high school students studying quadratic equations and discriminants.

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