Đề bài

Chứng minh các đẳng thức sau: a) \({\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab\); b) \({a^3} + {b^3} = \left( {a + b} \right)\left[ {{{\left( {a - b} \right)}^2} + ab} \right]\); c) \(2\left( {a - b} \right)\left( {a + b} \right) + {\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 4{a^2}\); d) \({\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac\).

Lời giải chi tiết

a) \({\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = {a^2} + 2ab + {b^2} - {a^2} + 2ab - {b^2}\)\( = \left( {{a^2} - {a^2}} \right) + \left( {2ab + 2ab} \right) + \left( {{b^2} - {b^2} = } \right)4ab\) (đpcm)b) \({a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) = \left( {a + b} \right)\left( {{a^2} - 2ab + {b^2} + ab} \right) = \left( {a + b} \right)\left[ {{{\left( {a - b} \right)}^2} + ab} \right]\)c) \(2\left( {a - b} \right)\left( {a + b} \right) + {\left( {a + b} \right)^2} + {\left( {a - b} \right)^2} = 2\left( {{a^2} - {b^2}} \right) + {a^2} + 2ab + {b^2} + {a^2} - 2ab + {b^2}\)\( = \left( {2{a^2} + {a^2} + {a^2}} \right) + \left( {{b^2} + {b^2} - 2{b^2}} \right) + \left( {2ab - 2ab} \right) = 4{a^2}\)d) \({\left( {a + b + c} \right)^2} = {\left[ {\left( {a + b} \right) + c} \right]^2} = {\left( {a + b} \right)^2} + 2\left( {a + b} \right)c + {c^2}\)\( = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac\)