For what values of k does the quadratic equation `4x^(2) - 12x - k =0 ` have no real roots ?

Answer

To determine the values of \( k \) for which the quadratic equation \( 4x^2 - 12x - k = 0 \) has no real roots, we need to analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 4 \)
- \( b = -12 \)
- \( c = -k \)

2. **Write the discriminant**:
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
For our equation, substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-12)^2 - 4 \cdot 4 \cdot (-k) \]

3. **Calculate the discriminant**:
Now, calculate \( D \):
\[ D = 144 + 16k \]

4. **Set the condition for no real roots**:
A quadratic equation has no real roots when the discriminant is less than zero:
\[ D < 0 \]
Therefore, we set up the inequality:
\[ 144 + 16k < 0 \]

5. **Solve the inequality**:
To find the values of \( k \), we first isolate \( k \):
\[ 16k < -144 \]
Now, divide both sides by 16:
\[ k < -\frac{144}{16} \]
Simplifying the fraction:
\[ k < -9 \]

### Conclusion:
The values of \( k \) for which the quadratic equation \( 4x^2 - 12x - k = 0 \) has no real roots are:
\[ k < -9 \]