If `x^(2) + kx + 3k = 0` has no solution , then the value of k will satisfy

Options

A. `k lt 12`
B. `k gt 12`
C. `k gt -12`
D. `0 lt k lt 12`

Answer

To determine the value of \( k \) for which the quadratic equation \( x^2 + kx + 3k = 0 \) has no solution, we need to analyze the discriminant of the quadratic equation. The discriminant is given by the formula:

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

For the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = k \)
- \( c = 3k \)

### Step 1: Calculate the Discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula, we get:

\[ D = k^2 - 4 \cdot 1 \cdot 3k \]

This simplifies to:

\[ D = k^2 - 12k \]

### Step 2: Set the Discriminant Less Than Zero
For the quadratic equation to have no solution, the discriminant must be less than zero:

\[ k^2 - 12k < 0 \]

### Step 3: Factor the Inequality
We can factor the left-hand side:

\[ k(k - 12) < 0 \]

### Step 4: Analyze the Inequality
Now, we need to determine the values of \( k \) that satisfy this inequality. The critical points are \( k = 0 \) and \( k = 12 \). We will test the intervals defined by these points:

1. **Interval 1:** \( k < 0 \)
- Choose \( k = -1 \): \( (-1)(-1 - 12) = (-1)(-13) = 13 > 0 \) (not valid)

2. **Interval 2:** \( 0 < k < 12 \)
- Choose \( k = 6 \): \( (6)(6 - 12) = (6)(-6) = -36 < 0 \) (valid)

3. **Interval 3:** \( k > 12 \)
- Choose \( k = 13 \): \( (13)(13 - 12) = (13)(1) = 13 > 0 \) (not valid)

### Conclusion
The values of \( k \) that satisfy the inequality \( k(k - 12) < 0 \) are:

\[ 0 < k < 12 \]

Thus, the value of \( k \) will satisfy:

\[ k \in (0, 12) \]