12/x-12+12/x-13=2

To solve the equation \( \frac{12}{x-12} + \frac{12}{x-13} = 2 \), we can follow these steps: 1. Obtain a common denominator for the two fractions on the left side. 2. Combine the fractions. 3. Solve the resulting equation. Step 1: The common denominator for \( x-12 \) and \( x-13 \) is \( (x-12)(x-13) \). Rewrite the equation with the common denominator: \( \frac{12(x-13)}{(x-12)(x-13)} + \frac{12(x-12)}{(x-12)(x-13)} = 2 \) Step 2: Combine the fractions on the left side: \( \frac{12(x-13) + 12(x-12)}{(x-12)(x-13)} = 2 \) \( \frac{12x - 156 + 12x - 144}{(x-12)(x-13)} = 2 \) \( \frac{24x - 300}{(x-12)(x-13)} = 2 \) Step 3: Now, simplify the equation: \( \frac{24x - 300}{(x-12)(x-13)} = 2 \) \( 24x - 300 = 2(x-12)(x-13) \) \( 24x - 300 = 2(x^2 - 25x + 156) \) \( 24x - 300 = 2x^2 - 50x + 312 \) \( 0 = 2x^2 - 74x + 612 \) Now, solve the quadratic equation: \( 2x^2 - 74x + 612 = 0 \) \( x^2 - 37x + 306 = 0 \) Factor the quadratic equation: \( (x-9)(x-28) = 0 \) Therefore, the solutions are \( x = 9 \) and \( x = 28 \).