Question

If a+b+c = 5 and ab+bc+ca = 10. prove that a3 + b3 + c3 -3abc=-25​

Answer 1

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$a + b + c = 5 \\ ab + bc + ca = 10 \\ (a + b + c) ^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ca) \\ {5}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2 \times 10 \\ {a}^{2} + {b}^{2} + {c}^{2} = 25 - 20 \\ {a}^{2} + {b}^{2} + {c}^{2} = 5 \\ \\ {a}^{3} + {b}^{3} + {c}^{3} - 3abc = (a + b + c)( {a}^{2} + {b}^{2} + {c}^{2} - ab - bc - ca) \\ = 5(5 - 10) \\ = - 25$