Solve `(-2)/(x^2-2)=(2)/(x-4)`

Options

A. (-6,2)
B. (-3,4)
C. (-3,2)
D. (-2,6)

Answer

To solve the equation \(\frac{-2}{x^2 - 2} = \frac{2}{x - 4}\), we will follow these steps:

### Step 1: Cross Multiply
We start by cross-multiplying the fractions to eliminate the denominators:
\[ -2 \cdot (x - 4) = 2 \cdot (x^2 - 2) \]

### Step 2: Distribute
Now, we will distribute both sides:
\[ -2x + 8 = 2x^2 - 4 \]

### Step 3: Rearrange the Equation
Next, we will rearrange the equation to bring all terms to one side:
\[ 2x^2 + 2x - 12 = 0 \]

### Step 4: Simplify
We can simplify the equation by dividing all terms by 2:
\[ x^2 + x - 6 = 0 \]

### Step 5: Factor the Quadratic
Now, we will factor the quadratic equation:
\[ (x + 3)(x - 2) = 0 \]

### Step 6: Solve for x
Setting each factor to zero gives us the possible solutions:
1. \(x + 3 = 0 \Rightarrow x = -3\)
2. \(x - 2 = 0 \Rightarrow x = 2\)

### Final Answer
Thus, the solutions to the equation are:
\[ x = -3 \quad \text{and} \quad x = 2 \]

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