Question

– Se m e n são as raízes de x2 – x – 30 = 0, calcule o valor da expressão 1
m
+
1
n
.

Answer 1

Resposta:

O valor da expressão é igual a $\mathbf{\dfrac{1}{m}+\dfrac{1}{n}=-\dfrac{1}{30}}$

Explicação passo-a-passo:

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Usando Bhaskara, temos

$x^2-x-30=0\\\\\\a=1\\b=-1\\c=-30\\\\\\\Delta=b^2-4\;.\;a\;.\;c=(-1)^2-4\;.\;1\;.\;-30=1+120=121\\\\\\x=\dfrac{-b \pm \sqrt{\Delta}}{2\;.\;a}=\dfrac{-(-1) \pm \sqrt{121}}{2\;.\;1}=\dfrac{1 \pm 11}{2}\\\\\\x_1=m=\dfrac{1 + 11}{2}=\dfrac{12}{2}=6\\\\\\x_2=n=\dfrac{1 - 11}{2}=\dfrac{-10}{2}=-5$

Portanto,

$\boxed{\dfrac{1}{m}+\dfrac{1}{n}=\dfrac{1}{6}-\dfrac{1}{5}=\dfrac{5-6}{30}=-\dfrac{1}{30}}$