a^2 /(a*b+b^2) + b^2 /(a*b-a^2) – (a^2 +b^2)/(a*b)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
\dfrac{{{a^2}}}{{ab + {b^2}}} + \dfrac{{{b^2}}}{{ab – {a^2}}} – \dfrac{{{a^2} + {b^2}}}{{ab}}\\
 = \dfrac{{{a^2}}}{{b\left( {a + b} \right)}} + \dfrac{{{b^2}}}{{a\left( {b – a} \right)}} – \dfrac{{{a^2} + {b^2}}}{{ab}}\\
 = \dfrac{{{a^2}}}{{b\left( {a + b} \right)}} – \dfrac{{{b^2}}}{{a\left( {a – b} \right)}} – \dfrac{{{a^2} + {b^2}}}{{ab}}\\
 = \dfrac{{{a^2}.a.\left( {a – b} \right) – {b^2}.b.\left( {a + b} \right) – \left( {{a^2} + {b^2}} \right)\left( {a + b} \right)\left( {a – b} \right)}}{{ab\left( {a – b} \right)\left( {a + b} \right)}}\\
 = \dfrac{{{a^3}\left( {a – b} \right) – {b^3}\left( {a + b} \right) – \left( {{a^2} + {b^2}} \right)\left( {{a^2} – {b^2}} \right)}}{{ab\left( {a – b} \right)\left( {a + b} \right)}}\\
 = \dfrac{{{a^4} – {a^3}b – {b^3}a – {b^4} – \left( {{a^4} – {b^4}} \right)}}{{ab\left( {a – b} \right)\left( {a + b} \right)}}\\
 = \dfrac{{{a^4} – {a^3}b – {b^3}a – {b^4} – {a^4} + {b^4}}}{{ab\left( {a – b} \right)\left( {a + b} \right)}}\\
 = \dfrac{{ – {a^3}b – a{b^3}}}{{ab\left( {a – b} \right)\left( {a + b} \right)}}\\
 = \dfrac{{ – ab\left( {{a^2} + {b^2}} \right)}}{{ab\left( {a – b} \right)\left( {a + b} \right)}}\\
 = \dfrac{{ – \left( {{a^2} + {b^2}} \right)}}{{{a^2} – {b^2}}}\\
 = \dfrac{{{a^2} + {b^2}}}{{{b^2} – {a^2}}}
\end{array}\)