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Sendo r1 e r2 as raizes da equação x(2)-x-5=0, calcule r1/r2 + r2/r1


RESOLVENDO

X²-x-5=0
Δ=1+20
Δ=21
√Δ= ± √21
\( x’= \frac{1+ \sqrt{21} }{2} \)
\( x"= \frac{1- \sqrt{21} }{2} \)
Calculando
\( \frac{ x^{’} }{ x^{"} } + \frac{ x^{"} }{ x^{’} } = \)
\( ( \frac{1+ \sqrt{21} }{2} : \frac{1- \sqrt{21} }{2} )+( \frac{1- \sqrt{21} }{2} : \frac{1+ \sqrt{21} }{2} )= \)
\( ( \frac{1+ \sqrt{21} }{2}. \frac{2}{1- \sqrt{21} } )+( \frac{1- \sqrt{21} }{2}. \frac{2}{1+ \sqrt{21} } )= \)
\( \frac{1+ \sqrt{21} }{1- \sqrt{21} } + \frac{1- \sqrt{21} }{1+ \sqrt{21} }= \)
\( \frac{(1+ \sqrt{21 } ) ^{2}+(1- \sqrt{21}) ^{2} }{(1- \sqrt{21})(1+ \sqrt{21} ) } = \)
\( \frac{1+2 \sqrt{21}+21+1-2 \sqrt{21}+21 }{1-21} = \)
\( = \frac{44}{-20} =- \frac{44}{20} =- \frac{11}{5} \)



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