If the equation `x^(2)+4x+k=0` has real and distinct roots, then find the value of 'k'.

Answer

To determine the value of \( k \) for which the quadratic equation \( x^2 + 4x + k = 0 \) has real and distinct roots, we need to use the discriminant condition.

### Step 1: Identify the coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = 4 \)
- \( c = k \)

### Step 2: Write the discriminant condition
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ D > 0 \]

### Step 3: Substitute the coefficients into the discriminant formula
Substituting \( a \), \( b \), and \( c \) into the discriminant formula, we get:
\[ D = 4^2 - 4 \cdot 1 \cdot k \]
This simplifies to:
\[ D = 16 - 4k \]

### Step 4: Set up the inequality
For the roots to be real and distinct, we need:
\[ 16 - 4k > 0 \]

### Step 5: Solve the inequality
Rearranging the inequality gives:
\[ 16 > 4k \]
Dividing both sides by 4:
\[ 4 > k \]
or
\[ k < 4 \]

### Conclusion
The value of \( k \) must be less than 4 for the quadratic equation \( x^2 + 4x + k = 0 \) to have real and distinct roots.