Question

Simplificar (m-x)²-p²/(x+p)²-m²

Answer 1

Olá!

$\\ \mathsf{\frac{(m - x)^2 - p^2}{(x + p)^2 - m^2} =} \\\\\\ \mathsf{\frac{\left [ (m - x) + p \right ] \cdot \left [ (m - x) - p \right ]}{\left [ (x + p) + m \right ] \cdot \left [ (x + p) - m \right ]} =} \\\\\\ \mathsf{\frac{(m - x + p)(m - x - p)}{(x + p + m)(x + p - m)} =} \\\\\\ \mathsf{\frac{(m - x + p)(m - x - p)}{(m + x + p)(- m + x + p)} =}$
 
 Note que os segundos fatores (tanto no numerador quanto no denominador) são simétricos. Daí,

$\\ \mathsf{\frac{(m - x + p) \cdot \left [ - 1 \cdot (- m + x + p) \right ]}{(m + x + p) \cdot (- m + x + p)} =} \\\\\\ \mathsf{- \frac{(m - x + p) \cdot (- m + x + p)}{(m + x + p) \cdot (- m + x + p)} =} \\\\\\ \boxed{\mathsf{- \frac{m - x + p}{m + x + p}}} \quad \mathsf{ou} \quad \boxed{\mathsf{\frac{- m + x - p}{m + x + p}}}$