The least value of `k` which makes the roots of the equation `x^2+5x+k=0` imaginary is

Options

A. 4
B. 5
C. 6
D. 7

Explanation

To find the least value of \( k \) that makes the roots of the equation \( x^2 + 5x + k = 0 \) imaginary, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 5 \)
- \( c = k \)

### Step 2: Use the condition for imaginary roots
The roots of a quadratic equation are imaginary when the discriminant \( D \) is less than zero. The discriminant is given by the formula:
\[ D = b^2 - 4ac \]
For our equation, we substitute the values of \( a \), \( b \), and \( c \):
\[ D = 5^2 - 4 \cdot 1 \cdot k \]
This simplifies to:
\[ D = 25 - 4k \]

### Step 3: Set up the inequality
For the roots to be imaginary, we need:
\[ D < 0 \]
Substituting the expression for \( D \):
\[ 25 - 4k < 0 \]

### Step 4: Solve the inequality
Rearranging the inequality gives:
\[ 25 < 4k \]
Dividing both sides by 4:
\[ \frac{25}{4} < k \]
This means:
\[ k > 6.25 \]

### Step 5: Determine the least integer value of \( k \)
Since we are looking for the least value of \( k \) that satisfies this inequality, we take the smallest integer greater than \( 6.25 \), which is:
\[ k = 7 \]

### Conclusion
The least value of \( k \) which makes the roots of the equation \( x^2 + 5x + k = 0 \) imaginary is:
\[ \boxed{7} \]