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 \( x = 28 \).