If `x ^(2) + 5x + 6 = 0,` then the value of `(2x )/( x ^(2) - 7x + 6)` is

Options

A. `1/6`
B. `1/3`
C. `-1/6`
D. `-1/3`

Answer

To solve the equation \( x^2 + 5x + 6 = 0 \) and find the value of \( \frac{2x}{x^2 - 7x + 6} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the equation:
\[ x^2 + 5x + 6 = 0 \]
We can factor this equation:
\[ (x + 2)(x + 3) = 0 \]
Setting each factor to zero gives us the solutions:
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

### Step 2: Substitute \( x \) into the expression
Next, we need to evaluate the expression \( \frac{2x}{x^2 - 7x + 6} \). We will calculate it for both values of \( x \).

#### For \( x = -2 \):
Substituting \( x = -2 \) into the denominator:
\[ x^2 - 7x + 6 = (-2)^2 - 7(-2) + 6 = 4 + 14 + 6 = 24 \]
Now substituting into the expression:
\[ \frac{2(-2)}{24} = \frac{-4}{24} = -\frac{1}{6} \]

#### For \( x = -3 \):
Substituting \( x = -3 \) into the denominator:
\[ x^2 - 7x + 6 = (-3)^2 - 7(-3) + 6 = 9 + 21 + 6 = 36 \]
Now substituting into the expression:
\[ \frac{2(-3)}{36} = \frac{-6}{36} = -\frac{1}{6} \]

### Conclusion
In both cases, the value of \( \frac{2x}{x^2 - 7x + 6} \) is:
\[ -\frac{1}{6} \]

### Final Answer
Thus, the answer is:
\[ \boxed{-\frac{1}{6}} \]