The value of k for which the equation `(k-2) x^(2) + 8x + k + 4 = 0` has both roots real, distinct and negative, is

Options

A. 0
B. 2
C. 3
D. 4

Answer

To find the value of \( k \) for which the equation \( (k-2)x^2 + 8x + (k+4) = 0 \) has both roots real, distinct, and negative, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be written in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = k - 2 \)
- \( b = 8 \)
- \( c = k + 4 \)

### Step 2: Condition for real and distinct roots
For the roots to be real and distinct, the discriminant \( D \) must be greater than 0:
\[ D = b^2 - 4ac > 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = 8^2 - 4(k - 2)(k + 4) > 0 \]
\[ D = 64 - 4[(k - 2)(k + 4)] > 0 \]

### Step 3: Expand the discriminant
Now, we expand the expression:
\[ (k - 2)(k + 4) = k^2 + 4k - 2k - 8 = k^2 + 2k - 8 \]
Substituting this back into the discriminant:
\[ D = 64 - 4(k^2 + 2k - 8) > 0 \]
\[ D = 64 - 4k^2 - 8k + 32 > 0 \]
\[ D = 96 - 4k^2 - 8k > 0 \]

### Step 4: Rearranging the inequality
Rearranging gives:
\[ 4k^2 + 8k - 96 < 0 \]
Dividing the entire inequality by 4:
\[ k^2 + 2k - 24 < 0 \]

### Step 5: Factor the quadratic
Now, we factor the quadratic:
\[ k^2 + 2k - 24 = (k - 4)(k + 6) \]
Thus, we need to solve:
\[ (k - 4)(k + 6) < 0 \]

### Step 6: Determine the intervals
The roots of the equation are \( k = 4 \) and \( k = -6 \). We analyze the sign of the product in the intervals:
- \( k < -6 \): Both factors are negative, product is positive.
- \( -6 < k < 4 \): One factor is negative and the other is positive, product is negative.
- \( k > 4 \): Both factors are positive, product is positive.

Thus, the solution to the inequality is:
\[ -6 < k < 4 \]

### Step 7: Condition for negative roots
Next, we need to ensure that both roots are negative. The roots of the quadratic equation can be found using the formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
For the roots to be negative, we need:
\[ \frac{-b}{a} > 0 \]
Substituting \( a \) and \( b \):
\[ \frac{-8}{k - 2} > 0 \]
This implies:
- If \( k - 2 > 0 \) (i.e., \( k > 2 \)), then \( -8 < 0 \) (which is true).
- If \( k - 2 < 0 \) (i.e., \( k < 2 \)), then \( -8 > 0 \) (which is false).

Thus, we need \( k > 2 \).

### Final Step: Combine the conditions
Combining the two conditions:
1. \( -6 < k < 4 \)
2. \( k > 2 \)

The valid range for \( k \) is:
\[ 2 < k < 4 \]

### Conclusion
The values of \( k \) for which the equation has both roots real, distinct, and negative are:
\[ k \in (2, 4) \]