If `x^(2)+ kx+7.5k =0` has no solution, them the value of k will satisfy:

Options

A. `0lt k lt 30`
B. `k lt 0`
C. `k gt -30`
D. `k lt 30`

Answer

To determine the values of \( k \) for which the quadratic equation \( x^2 + kx + 7.5k = 0 \) has no solutions, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) is given by the formula:

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

where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Identify coefficients
For the given equation \( x^2 + kx + 7.5k = 0 \):
- \( a = 1 \)
- \( b = k \)
- \( c = 7.5k \)

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

\[ D = k^2 - 4(1)(7.5k) \]

### Step 3: Simplify the discriminant
Now, simplify the expression:

\[ D = k^2 - 30k \]

### Step 4: Set the condition for no solutions
For the quadratic equation to have no solutions, the discriminant must be less than zero:

\[ k^2 - 30k < 0 \]

### Step 5: Factor the inequality
We can factor the left-hand side:

\[ k(k - 30) < 0 \]

### Step 6: Analyze the inequality
To solve the inequality \( k(k - 30) < 0 \), we need to find the intervals where the product is negative. The roots of the equation \( k(k - 30) = 0 \) are \( k = 0 \) and \( k = 30 \).

### Step 7: Test intervals
We will test the intervals determined by the roots:
1. \( k < 0 \)
2. \( 0 < k < 30 \)
3. \( k > 30 \)

- For \( k < 0 \): Both factors \( k \) and \( (k - 30) \) are negative, so the product is positive.
- For \( 0 < k < 30 \): The factor \( k \) is positive and \( (k - 30) \) is negative, so the product is negative.
- For \( k > 30 \): Both factors are positive, so the product is positive.

### Step 8: Conclusion
The inequality \( k(k - 30) < 0 \) holds true in the interval:

\[ 0 < k < 30 \]

Thus, the value of \( k \) that satisfies the condition for the quadratic equation to have no solutions is:

\[ \boxed{(0, 30)} \]