Question

Por favor ayúdenme con este ejercicio

Answer 1

° Enunciado:
$\frac{x}{(x - 2)(x - 3)} = ( \frac{1}{ {x}^{2} - x - 2} + \frac{1}{ {x}^{2} - 5x + 6} - \frac{ 1}{ {x}^{2} - 2x - 3 } )$

° Factorizamos:
$\frac{x}{(x - 2)(x - 3)} = ( \frac{1}{ {x}^{2} - x - 2} + \frac{1}{ {x}^{2} - 5x + 6} - \frac{ 1}{ {x}^{2} - 2x - 3 } ) \\ \\ \frac{x}{(x - 2)(x - 3)} = \frac{1}{(x - 2)(x +1 )} + \frac{1}{(x - 3)(x - 2)} - \frac{1}{(x -3 )(x + 1)} \\ \\ \frac{x}{(x - 2)(x - 3)} = \frac{(x - 3) + (x + 1) - (x - 2)}{(x - 2)(x - 3)(x + 1)} \\ \\ \frac{x}{(x - 2)(x - 3)} = \frac{x - 3+ x + 1 - x + 2)}{(x - 2)(x - 3)(x + 1)} \\ \\ \frac{x}{(x - 2)(x - 3)} = \frac{x }{(x - 2)(x - 3)(x + 1)} \\ \\ x = \frac{x }{(x - 2)(x - 3)(x + 1)}.(x - 2)(x - 3) \\ \\ x = \frac{x}{x + 1} \\ \\ x(x + 1) = x \\ \\ {x}^{2} + x = x \\ \\ {x}^{2} + x - x = 0 \\ \\ {x}^{2} = 0 \\ \\ x = \sqrt{0} \\ \\ \boxed{ x = 0}$