Question

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

Answer 1

x²-x-5=0
Δ=1+20
Δ=21
√Δ= ± √21

$x'= \frac{1+ \sqrt{21} }{2}$

$x"= \frac{1- \sqrt{21} }{2}$

Calculando
$\frac{ x^{'} }{ x^{"} } + \frac{ x^{"} }{ x^{'} } =$

$( \frac{1+ \sqrt{21} }{2} : \frac{1- \sqrt{21} }{2} )+( \frac{1- \sqrt{21} }{2} : \frac{1+ \sqrt{21} }{2} )=$

$( \frac{1+ \sqrt{21} }{2} . \frac{2}{1- \sqrt{21} } )+( \frac{1- \sqrt{21} }{2} . \frac{2}{1+ \sqrt{21} } )=$

$\frac{1+ \sqrt{21} }{1- \sqrt{21} } + \frac{1- \sqrt{21} }{1+ \sqrt{21} }=$

$\frac{(1+ \sqrt{21 } ) ^{2}+(1- \sqrt{21}) ^{2} }{(1- \sqrt{21})(1+ \sqrt{21} ) } =$

$\frac{1+2 \sqrt{21}+21+1-2 \sqrt{21}+21 }{1-21} =$

$= \frac{44}{-20} =- \frac{44}{20} =- \frac{11}{5}$