For what values of k are the roots of the equation `kx^(2)+4x+k=0` real and unequal ?

Options

A. `0 lt k lt 2`
B. `|k| lt 2 `
C. `|k| gt 2`
D. `-2 lt k lt 0 ` or `0 lt k lt 2`

Explanation

To determine the values of \( k \) for which the roots of the equation \( kx^2 + 4x + k = 0 \) are real and unequal, we will use the discriminant condition. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where the discriminant \( D \) is given by:

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

For the roots to be real and unequal, the discriminant must be greater than zero:

\[ D > 0 \]

### Step 1: Identify \( a \), \( b \), and \( c \)
From the equation \( kx^2 + 4x + k = 0 \), we identify:
- \( a = k \)
- \( b = 4 \)
- \( c = k \)

### Step 2: Write the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:

\[ D = b^2 - 4ac = 4^2 - 4(k)(k) \]

### Step 3: Simplify the discriminant
Calculating \( D \):

\[ D = 16 - 4k^2 \]

### Step 4: Set the discriminant greater than zero
To ensure the roots are real and unequal, we set up the inequality:

\[ 16 - 4k^2 > 0 \]

### Step 5: Solve the inequality
Rearranging the inequality gives:

\[ 16 > 4k^2 \]

Dividing both sides by 4:

\[ 4 > k^2 \]

### Step 6: Take the square root
Taking the square root of both sides gives:

\[ 2 > |k| \]

This can be expressed as:

\[ -2 < k < 2 \]

### Step 7: Consider the case when \( k \neq 0 \)
Since \( k \) is the coefficient of \( x^2 \), it cannot be equal to zero (as it would not be a quadratic equation). Therefore, we exclude \( k = 0 \).

### Step 8: Final intervals for \( k \)
The valid intervals for \( k \) are:

\[ k \in (-2, 0) \cup (0, 2) \]

### Conclusion
Thus, the values of \( k \) for which the roots of the equation \( kx^2 + 4x + k = 0 \) are real and unequal are:

\[ k \in (-2, 0) \cup (0, 2) \]