`x^(3)+5x^(2)+px+q=0` and `x^(3)+7x^(2)+px+r=0` have two roos in common. If their third roots are `gamma_(1)` and `gamma_(2)` , respectively, then `|gamma_(1)-gamma_(2)|=`

Options

A. `10`
B. `12`
C. `13`
D. `42`

Explanation

To solve the problem, we need to analyze the two cubic equations given and find the absolute difference between their third roots, denoted as \( \gamma_1 \) and \( \gamma_2 \).

### Step-by-Step Solution:

1. **Identify the Equations:**
We have two cubic equations:
\[ x^3 + 5x^2 + px + q = 0 \quad (1) \]
\[ x^3 + 7x^2 + px + r = 0 \quad (2) \]
The equations have two roots in common, which we will denote as \( \alpha \) and \( \beta \). The third roots are \( \gamma_1 \) for equation (1) and \( \gamma_2 \) for equation (2).

2. **Use Vieta's Formulas:**
According to Vieta's formulas, for a cubic equation \( x^3 + ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -a \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \alpha\gamma = b \)
- The product of the roots \( \alpha\beta\gamma = -c \)

For equation (1):
\[ \alpha + \beta + \gamma_1 = -5 \quad (3) \]

For equation (2):
\[ \alpha + \beta + \gamma_2 = -7 \quad (4) \]

3. **Subtract Equations (3) and (4):**
By subtracting equation (4) from equation (3):
\[ (\alpha + \beta + \gamma_1) - (\alpha + \beta + \gamma_2) = -5 + 7 \]
This simplifies to:
\[ \gamma_1 - \gamma_2 = 2 \quad (5) \]

4. **Find the Absolute Difference:**
We need to find \( |\gamma_1 - \gamma_2| \):
\[ |\gamma_1 - \gamma_2| = |2| = 2 \]

### Final Answer:
\[ |\gamma_1 - \gamma_2| = 2 \]