If product of two roots of ` 4x^3 + 16 x^2 +kx - 36 =0` is -6 then k=

Options

A. 9
B. `-9`
C. `8`
D. `-8`

Answer

To solve the problem, we need to find the value of \( k \) in the cubic equation \( 4x^3 + 16x^2 + kx - 36 = 0 \) given that the product of two of its roots is -6.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the equation be \( \alpha, \beta, \gamma \). According to the problem, we know that:
\[ \alpha \beta = -6 \]

2. **Use Vieta's Formulas**:
From Vieta's formulas for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{b}{a} = -\frac{16}{4} = -4 \) (Equation 1)
- The sum of the product of the roots taken two at a time \( \alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} = \frac{k}{4} \) (Equation 2)
- The product of the roots \( \alpha \beta \gamma = -\frac{d}{a} = -\frac{-36}{4} = 9 \) (Equation 3)

3. **Substituting the Known Product**:
From Equation 3, we have:
\[ \alpha \beta \gamma = 9 \]
Since \( \alpha \beta = -6 \), we can express \( \gamma \) as:
\[ \gamma = \frac{9}{\alpha \beta} = \frac{9}{-6} = -\frac{3}{2} \]

4. **Finding \( \alpha + \beta \)**:
Using Equation 1:
\[ \alpha + \beta + \gamma = -4 \]
Substituting \( \gamma = -\frac{3}{2} \):
\[ \alpha + \beta - \frac{3}{2} = -4 \]
Rearranging gives:
\[ \alpha + \beta = -4 + \frac{3}{2} = -\frac{8}{2} + \frac{3}{2} = -\frac{5}{2} \]

5. **Substituting into Equation 2**:
Now we substitute \( \alpha \beta = -6 \) and \( \alpha + \beta = -\frac{5}{2} \) into Equation 2:
\[ \alpha \beta + \beta \gamma + \gamma \alpha = \frac{k}{4} \]
Substituting the known values:
\[ -6 + \gamma(\alpha + \beta) = \frac{k}{4} \]
Substituting \( \gamma = -\frac{3}{2} \) and \( \alpha + \beta = -\frac{5}{2} \):
\[ -6 + \left(-\frac{3}{2}\right)\left(-\frac{5}{2}\right) = \frac{k}{4} \]
Calculating \( \left(-\frac{3}{2}\right)\left(-\frac{5}{2}\right) = \frac{15}{4} \):
\[ -6 + \frac{15}{4} = \frac{k}{4} \]
Converting -6 to a fraction:
\[ -6 = -\frac{24}{4} \]
Thus:
\[ -\frac{24}{4} + \frac{15}{4} = \frac{k}{4} \]
Simplifying gives:
\[ -\frac{9}{4} = \frac{k}{4} \]
Multiplying both sides by 4:
\[ k = -9 \]

### Final Answer:
Thus, the value of \( k \) is \( \boxed{-9} \).