`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 step by step, we start by analyzing the given cubic equations and their roots.

### Step 1: Define the roots of the equations
Let the roots of the first equation \( x^3 + 5x^2 + px + q = 0 \) be \( \alpha_1, \beta_1, \gamma_1 \).
Let the roots of the second equation \( x^3 + 7x^2 + px + r = 0 \) be \( \alpha_1, \beta_1, \gamma_2 \).
Here, \( \alpha_1 \) and \( \beta_1 \) are the common roots, and \( \gamma_1 \) and \( \gamma_2 \) are the third roots of the respective equations.

### Step 2: Use Vieta's formulas
From Vieta's formulas, we know:
- For the first equation:
\[ \alpha_1 + \beta_1 + \gamma_1 = -5 \]
- For the second equation:
\[ \alpha_1 + \beta_1 + \gamma_2 = -7 \]

### Step 3: Express \( \gamma_1 \) and \( \gamma_2 \) in terms of \( \alpha_1 \) and \( \beta_1 \)
Since \( \alpha_1 + \beta_1 \) is common in both equations, we can denote:
\[ S = \alpha_1 + \beta_1 \]
Thus, we can rewrite the equations as:
- From the first equation:
\[ S + \gamma_1 = -5 \implies \gamma_1 = -5 - S \]
- From the second equation:
\[ S + \gamma_2 = -7 \implies \gamma_2 = -7 - S \]

### Step 4: Find \( \gamma_1 + \gamma_2 \)
Now, we can find \( \gamma_1 + \gamma_2 \):
\[ \gamma_1 + \gamma_2 = (-5 - S) + (-7 - S) = -12 - 2S \]

### Step 5: Find \( S \)
From the first equation, we know:
\[ S = \alpha_1 + \beta_1 \]
Since \( S \) is not given directly, we can use the fact that the sum of the roots \( \alpha_1 + \beta_1 \) must equal \( 0 \) (as derived from the subtraction of the two equations). Therefore, we can conclude:
\[ S = 0 \]

### Step 6: Substitute \( S \) back into the equation for \( \gamma_1 + \gamma_2 \)
Substituting \( S = 0 \) into the equation for \( \gamma_1 + \gamma_2 \):
\[ \gamma_1 + \gamma_2 = -12 - 2(0) = -12 \]

### Step 7: Find the absolute value
Finally, we need to find the absolute value:
\[ |\gamma_1 + \gamma_2| = |-12| = 12 \]

Thus, the final answer is:
\[ \boxed{12} \]