a^2-(b-c)^ 2/(a+c) ^ 2 -b ^ 2 + b ^ 2 -(a-c)^ 2/(a+b)^ 2 -c ^ 2 + c ^ 2 - (a-b)^ 2/(b+c)^ 2 - a^ 2
You asked:
MathBot Answer:
Evaluated
\(\displaystyle {a}^{2} - \frac{{\left( b - c \right)}^{2}}{{\left( a + c \right)}^{2}} - {b}^{2} + {b}^{2} - \frac{{\left( a - c \right)}^{2}}{{\left( a + b \right)}^{2}} - {c}^{2} + {c}^{2} - \frac{{\left( a - b \right)}^{2}}{{\left( b + c \right)}^{2}} - {a}^{2} = - \frac{\left(a - b\right)^{2}}{\left(b + c\right)^{2}} - \frac{\left(a - c\right)^{2}}{\left(a + b\right)^{2}} - \frac{\left(b - c\right)^{2}}{\left(a + c\right)^{2}} \)
Expanded
\[{a}^{2} - \frac{{\left( b - c \right)}^{2}}{{\left( a + c \right)}^{2}} - {b}^{2} + {b}^{2} - \frac{{\left( a - c \right)}^{2}}{{\left( a + b \right)}^{2}} - {c}^{2} + {c}^{2} - \frac{{\left( a - b \right)}^{2}}{{\left( b + c \right)}^{2}} - {a}^{2} = - \frac{a^{2}}{b^{2} + 2 b c + c^{2}} - \frac{a^{2}}{a^{2} + 2 a b + b^{2}} + \frac{2 a b}{b^{2} + 2 b c + c^{2}} + \frac{2 a c}{a^{2} + 2 a b + b^{2}} - \frac{b^{2}}{b^{2} + 2 b c + c^{2}} - \frac{b^{2}}{a^{2} + 2 a c + c^{2}} + \frac{2 b c}{a^{2} + 2 a c + c^{2}} - \frac{c^{2}}{a^{2} + 2 a c + c^{2}} - \frac{c^{2}}{a^{2} + 2 a b + b^{2}}\]
Factored
\[{a}^{2} - \frac{{\left( b - c \right)}^{2}}{{\left( a + c \right)}^{2}} - {b}^{2} + {b}^{2} - \frac{{\left( a - c \right)}^{2}}{{\left( a + b \right)}^{2}} - {c}^{2} + {c}^{2} - \frac{{\left( a - b \right)}^{2}}{{\left( b + c \right)}^{2}} - {a}^{2} = - \frac{a^{6} + 2 a^{5} c - a^{4} b^{2} + 2 a^{4} b c + 2 a^{4} c^{2} - 4 a^{3} b^{2} c + 2 a^{2} b^{4} - 6 a^{2} b^{2} c^{2} - 4 a^{2} c^{3} b - a^{2} c^{4} + 2 b^{5} a + 2 b^{4} a c - 4 b^{3} c^{2} a + 2 c^{4} a b + b^{6} - b^{4} c^{2} + 2 b^{2} c^{4} + 2 c^{5} b + c^{6}}{\left(a + b\right)^{2} \left(a + c\right)^{2} \left(b + c\right)^{2}}\]