if k is a positive integer, find the maximum value of k such that (k+8)x² + (k-2)x-3=0 has real roots

To find the maximum value of \( k \) such that the quadratic equation 

\[
(k+8)x^2 + (k-2)x - 3 = 0
\]

has real roots, we need to ensure that the discriminant of the equation is non-negative. 

### Step 1: Identify the Discriminant Condition for Real Roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by:

\[
\Delta = b^2 - 4ac
\]

The roots are real if and only if \( \Delta \geq 0 \).

### Step 2: Define \( a \), \( b \), and \( c \)
In our equation:
- \( a = k + 8 \)
- \( b = k - 2 \)
- \( c = -3 \)

### Step 3: Compute the Discriminant
Substitute these values into the discriminant formula:

\[
\Delta = (k - 2)^2 - 4(k + 8)(-3)
\]

Expanding this:

\[
\Delta = (k - 2)^2 + 12(k + 8)
\]

Calculate each term:

1. \((k - 2)^2 = k^2 - 4k + 4\)
2. \(12(k + 8) = 12k + 96\)

So,

\[
\Delta = k^2 - 4k + 4 + 12k + 96
\]

Combine like terms:

\[
\Delta = k^2 + 8k + 100
\]

### Step 4: Ensure the Discriminant is Non-negative
For the quadratic equation to have real roots, we need:

\[
k^2 + 8k + 100 \geq 0
\]

### Step 5: Analyze the Quadratic Expression
The expression \( k^2 + 8k + 100 \) is always positive for all values of \( k \) since its discriminant (\(8^2 - 4 \cdot 1 \cdot 100\)) is negative, indicating it has no real roots and opens upwards.

Therefore, no upper limit on \( k \) is needed for real roots to exist. The value of \( k \) can be any positive integer.

### Conclusion
The maximum value of \( k \) such that \((k+8)x^2 + (k-2)x - 3 = 0\) has real roots is **unbounded**, meaning \( k \) can be any positive integer.

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Would you like further clarification on any part of the solution?

### Related Questions:
1. What if \( k \) were negative? How would the result change?
2. How would we find the roots if \( k \) were a specific value, like 1 or 2?
3. Could there be any conditions on \( k \) for complex roots to occur?
4. What if we were asked for integer solutions for \( x \) given this quadratic?
5. How would this change if we maximized \( k \) for complex roots instead?

**Tip:** For any quadratic equation, always check the discriminant to determine the nature of its roots.

If k is a positive integer, find the maximum value of k such that (k+8)x² + (k-2)x - 3 = 0 has real roots.

Explore how to determine the maximum value of k such that (k+8)x² + (k-2)x - 3 = 0 has real roots. This problem involves quadratic equations, the discriminant formula, and the real roots theorem, making it ideal for students in Grades 10-12.

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