Question

A soma dos possíveis valores de x na equação é: 25x+5/6-5x=0 é:

Escolha uma:
a. 2
b. 1
c. 4
d. 0
e. 3

Answer 1

Interpretando a risca a equação seria:
$\mathsf{25x+\dfrac{5}{6}-5x=0}$
Logo:
$\mathsf{25x-5x=-\dfrac{5}{6}}\\\\\mathsf{20x=-\dfrac{5}{6}}\\\\\mathsf{x=-\dfrac{5}{6}\cdot \dfrac{1}{20}}\\\\\mathsf{x=-\dfrac{1}{6\cdot 4}}\\\\\mathsf{x=-\dfrac{1}{24}}$
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Outra interpretação:
$\mathsf{25x+\dfrac{5}{6-5x}=0}$
Resolvendo teremos:
$\mathsf{\dfrac{25x\cdot \left(6-5x\right)}{\left6-5x\right}+\dfrac{5}{6-5x}=0}\\\\\mathsf{\dfrac{150x-125x^2+5}{6-5x}=0}\\\\\mathsf{150x-125x^2+5=0}\\\\\mathsf{30x-25x^2+1=0}\\\\\mathsf{-25x^2+30+1=0}$
$\mathsf{a=-25\:,\:b=30\:,\:c=1}$

$\mathsf{x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}}\\\\\\\mathsf{x=\dfrac{-30\pm \sqrt{30^2-4\cdot \left(-25\right)\cdot 1}}{2\cdot \left(-25\right)}}\\\\\\\mathsf{x=\dfrac{-30\pm \sqrt{900+100}}{-50}}\\\\\\\mathsf{x=\dfrac{-30\pm \sqrt{1000}}{-50}}\\\\\\\mathsf{x=\dfrac{-30\pm 10\sqrt{10}}{-50}}\\\\\\\mathsf{x=\dfrac{-3\pm \sqrt{10}}{-5}}\\\\\\\mathsf{x=\dfrac{3\pm \sqrt{10}}{5}}$

As raizes são:
$\mathsf{\quad x_1=\dfrac{3+\sqrt{10}}{5}\:\:\:\:\:\:e\:\:\:\:\:\:x_2=\dfrac{3-\sqrt{10}}{5}}$
Logo, o somatória das raizes sera:
$\mathsf{S_{raizes}=\dfrac{3+\sqrt{10}}{5}+\dfrac{3-\sqrt{10}}{5}}\\\\\mathsf{S_{raizes}=\dfrac{3+\sqrt{10}+3-\sqrt{10}}{5}}\\\\\mathsf{S_{raizes}=\dfrac{3+3}{5}}\\\\\boxed{\mathsf{S_{raizes}=\dfrac{6}{5}}}\: \: \checkmark$