What is the smallest integral value oof k for which the roots `3x^(2)+8x-k=0` are real?

Options

A. `-6`
B. `-5`
C. `0`
D. `6`

Answer

To determine the smallest integral value of \( k \) for which the roots of the quadratic equation \( 3x^2 + 8x - k = 0 \) are real, 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 form \( ax^2 + bx + c = 0 \), where:
- \( a = 3 \)
- \( b = 8 \)
- \( c = -k \)

2. **Write the condition for real roots**:
For the roots of a quadratic equation to be real, the discriminant \( D \) must be greater than or equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
Therefore, we need:
\[ D \geq 0 \]

3. **Substitute the values of \( a \), \( b \), and \( c \)**:
Substituting the values into the discriminant formula, we get:
\[ D = 8^2 - 4 \cdot 3 \cdot (-k) \]

4. **Calculate the discriminant**:
\[ D = 64 + 12k \]

5. **Set up the inequality**:
To ensure the roots are real, we set up the inequality:
\[ 64 + 12k \geq 0 \]

6. **Solve for \( k \)**:
Rearranging the inequality gives:
\[ 12k \geq -64 \]
Dividing both sides by 12:
\[ k \geq -\frac{64}{12} \]
Simplifying the fraction:
\[ k \geq -\frac{16}{3} \approx -5.33 \]

7. **Determine the smallest integral value of \( k \)**:
Since \( k \) must be an integer, the smallest integer greater than or equal to \(-5.33\) is:
\[ k = -5 \]

### Conclusion:
The smallest integral value of \( k \) for which the roots of the equation \( 3x^2 + 8x - k = 0 \) are real is:
\[ \boxed{-5} \]