If the ratio of roots of equation `3x^(2)- kx+14= 0` is `7:6`, then value of k is

Options

A. 1
B. 12
C. 13
D. `-3`

Answer

To solve the problem, we need to find the value of \( k \) in the quadratic equation \( 3x^2 - kx + 14 = 0 \) given that the ratio of the roots is \( 7:6 \).

### Step 1: Use the ratio of the roots
Let the roots of the equation be \( 7x \) and \( 6x \) for some \( x \).

### Step 2: Apply Vieta's formulas
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)

In our case:
- \( r_1 + r_2 = 7x + 6x = 13x \)
- \( r_1 \cdot r_2 = 7x \cdot 6x = 42x^2 \)

### Step 3: Set up equations using Vieta's formulas
From Vieta's formulas:
1. The sum of the roots gives us:
\[ 13x = \frac{k}{3} \quad \text{(since } b = -k \text{ and } a = 3\text{)} \]
Therefore, we can express \( k \) as:
\[ k = 39x \]

2. The product of the roots gives us:
\[ 42x^2 = \frac{14}{3} \quad \text{(since } c = 14 \text{ and } a = 3\text{)} \]
Rearranging this gives:
\[ 42x^2 = \frac{14}{3} \implies x^2 = \frac{14}{3 \cdot 42} = \frac{14}{126} = \frac{1}{9} \]
Thus, we find:
\[ x = \frac{1}{3} \]

### Step 4: Substitute \( x \) back into the equation for \( k \)
Now substituting \( x = \frac{1}{3} \) back into the equation for \( k \):
\[ k = 39 \cdot \frac{1}{3} = 13 \]

### Final Answer
Thus, the value of \( k \) is \( \boxed{13} \).