Toán Cho a=b+c. Chứng minh: a^3+b^3/a^3+c^3 = a+b/a+c 09/09/2021 By Margaret Cho a=b+c. Chứng minh: a^3+b^3/a^3+c^3 = a+b/a+c
\[\begin{array}{l} \frac{{{a^3} + {b^3}}}{{{a^3} + {c^3}}} = \frac{{a + b}}{{a + c}}\\ \Leftrightarrow \frac{{\left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)}}{{\left( {a + c} \right)\left( {{a^2} – ac + {c^2}} \right)}} = \frac{{a + b}}{{a + c}}\\ \Leftrightarrow \frac{{{a^2} – ab + {b^2}}}{{{a^2} – ac + {c^2}}} = 1\\ \Leftrightarrow {a^2} – ab + {b^2} = {a^2} – ac + {c^2}\\ \Leftrightarrow {b^2} – ab = {c^2} – ac\\ \Leftrightarrow {b^2} – {c^2} + ac – ab = 0\\ \Leftrightarrow \left( {b – c} \right)\left( {b + c} \right) – a\left( {b – c} \right) = 0\\ \Leftrightarrow \left( {b – c} \right)\left( {b + c – a} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} b = c\\ a = b + c \end{array} \right. \end{array}\] Trả lời
\[\begin{array}{l}
\frac{{{a^3} + {b^3}}}{{{a^3} + {c^3}}} = \frac{{a + b}}{{a + c}}\\
\Leftrightarrow \frac{{\left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)}}{{\left( {a + c} \right)\left( {{a^2} – ac + {c^2}} \right)}} = \frac{{a + b}}{{a + c}}\\
\Leftrightarrow \frac{{{a^2} – ab + {b^2}}}{{{a^2} – ac + {c^2}}} = 1\\
\Leftrightarrow {a^2} – ab + {b^2} = {a^2} – ac + {c^2}\\
\Leftrightarrow {b^2} – ab = {c^2} – ac\\
\Leftrightarrow {b^2} – {c^2} + ac – ab = 0\\
\Leftrightarrow \left( {b – c} \right)\left( {b + c} \right) – a\left( {b – c} \right) = 0\\
\Leftrightarrow \left( {b – c} \right)\left( {b + c – a} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
b = c\\
a = b + c
\end{array} \right.
\end{array}\]