If one root of `x^2+kx +12 =0` may be the triple the other , then k=

Options

A. `+-8`
B. 3
C. `+-5sqrt(10)`
D. `+-2sqrt5`

Answer

To solve the problem, we need to find the value of \( k \) in the quadratic equation \( x^2 + kx + 12 = 0 \) given that one root is triple the other. Let's denote the roots of the equation as \( \alpha \) and \( 3\alpha \).

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots be \( \alpha \) and \( 3\alpha \).

2. **Sum of the Roots**:
According to Vieta's formulas, the sum of the roots is given by:
\[ \alpha + 3\alpha = -\frac{b}{a} = -\frac{k}{1} = -k \]
Simplifying this, we have:
\[ 4\alpha = -k \quad \text{(1)} \]

3. **Product of the Roots**:
The product of the roots is also given by Vieta's formulas:
\[ \alpha \cdot 3\alpha = \frac{c}{a} = \frac{12}{1} = 12 \]
This simplifies to:
\[ 3\alpha^2 = 12 \]
Dividing both sides by 3 gives:
\[ \alpha^2 = 4 \quad \text{(2)} \]

4. **Substituting for \( \alpha \)**:
From equation (2), we can find \( \alpha \):
\[ \alpha = \sqrt{4} = 2 \quad \text{or} \quad \alpha = -\sqrt{4} = -2 \]

5. **Finding \( k \)**:
Now, substituting \( \alpha = 2 \) into equation (1):
\[ 4(2) = -k \implies 8 = -k \implies k = -8 \]
If \( \alpha = -2 \):
\[ 4(-2) = -k \implies -8 = -k \implies k = 8 \]

6. **Final Values of \( k \)**:
Thus, the possible values for \( k \) are \( k = 8 \) or \( k = -8 \).

7. **Conclusion**:
Therefore, the final answer is:
\[ k = \pm 8 \]