Solve the following equation. \( \frac{1}{x-2}-\frac{2}{x^{2}-2 x}=4 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \{\square \). (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is \( \varnothing \).
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To solve the equation \( \frac{1}{x-2}-\frac{2}{x^{2}-2 x}=4 \), we first recognize that \( x^{2}-2x \) can be factored as \( x(x-2) \). Thus, rewriting our equation gives us: \[ \frac{1}{x-2} - \frac{2}{x(x-2)} = 4. \] Next, we can combine the fractions on the left-hand side using a common denominator: \[ \frac{x - 2}{x(x-2)} - \frac{2}{x(x-2)} = \frac{x - 4}{x(x-2)} = 4. \] Now we can multiply both sides by \( x(x-2) \) (assuming \( x \neq 0 \) and \( x \neq 2 \) to avoid division by zero): \[ x - 4 = 4x(x - 2). \] Rearranging gives us: \[ x - 4 = 4x^2 - 8x, \] \[ 0 = 4x^2 - 9x + 4. \] Now we can use the quadratic formula where \( a = 4, b = -9, c = 4 \): \[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \] \[ x = \frac{9 \pm \sqrt{81 - 64}}{8} \] \[ x = \frac{9 \pm \sqrt{17}}{8}. \] Thus, our solutions are: \[ x = \frac{9 + \sqrt{17}}{8}, \quad x = \frac{9 - \sqrt{17}}{8}. \] Both of these values are valid since neither equals 0 nor 2. Hence, the solution set is: **A. The solution set is \( \{\frac{9 + \sqrt{17}}{8}, \frac{9 - \sqrt{17}}{8}\} \)**
