Resposta:
[tex]\textsf{Leia abaixo}[/tex]
Explicação passo a passo:
[tex]\begin{cases}\mathsf{7\sqrt[3]{\textsf{xy}} - 3(xy)^{\frac{1}{2}} = 4}\\\mathsf{x + y = 20}\end{cases}[/tex]
[tex]\mathsf{7\sqrt[3]{\textsf{xy}} - 3\left(\sqrt[3]{\textsf{xy}}\right)^{\frac{3}{2}} = 4}[/tex]
[tex]\mathsf{k = \sqrt[3]{\textsf{xy}} }[/tex]
[tex]\mathsf{7k - 3k^{\frac{3}{2}} = 4}[/tex]
[tex]\mathsf{7k - 4 = 3k^{\frac{3}{2}}}[/tex]
[tex]\mathsf{(7k - 4)^2 = (3k^{\frac{3}{2}})^2}[/tex]
[tex]\mathsf{49k^2 - 56k + 16 = 9k^3}[/tex]
[tex]\mathsf{9k^3 - 49k^2 + 56k - 16 = 0}[/tex]
[tex]\mathsf{9k^2 - 40k + 16 = 0}[/tex]
[tex]\mathsf{\Delta = b^2 - 4.a.c}[/tex]
[tex]\mathsf{\Delta = (-40)^2 - 4.9.16}[/tex]
[tex]\mathsf{\Delta = 1600 - 576}[/tex]
[tex]\mathsf{\Delta = 1024}[/tex]
[tex]\mathsf{k = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{40 \pm \sqrt{1024}}{18} \rightarrow \begin{cases}\mathsf{k' = \dfrac{40 + 32}{18} = \dfrac{72}{18} = 4}\\\\\mathsf{k'' = \dfrac{40 - 32}{18} = \dfrac{8}{18} = \dfrac{4}{9}}\end{cases}}[/tex]
[tex]\mathsf{\sqrt[3]{\textsf{xy}} = 4}[/tex]
[tex]\mathsf{xy = 4^3}[/tex]
[tex]\mathsf{xy = 64}[/tex]
[tex]\mathsf{x = \dfrac{64}{y}}[/tex]
[tex]\mathsf{\dfrac{64}{y} + y = 20}[/tex]
[tex]\mathsf{64 + y^2 = 20y}[/tex]
[tex]\mathsf{y^2 - 20y + 64 = 0}[/tex]
[tex]\mathsf{\Delta = b^2 - 4.a.c}[/tex]
[tex]\mathsf{\Delta = (-20)^2 - 4.1.64}[/tex]
[tex]\mathsf{\Delta = 400 - 256}[/tex]
[tex]\mathsf{\Delta = 144}[/tex]
[tex]\mathsf{y = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{20 \pm \sqrt{144}}{2} \rightarrow \begin{cases}\mathsf{y' = \dfrac{20 + 12}{2} = \dfrac{32}{2} = 16}\\\\\mathsf{y'' = \dfrac{20 - 12}{2} = \dfrac{8}{2} = 4}\end{cases}}[/tex]
[tex]\mathsf{x' = \dfrac{64}{16} = 4}[/tex]
[tex]\mathsf{x'' = \dfrac{64}{4} = 16}[/tex]
[tex]\boxed{\boxed{\mathsf{S = \{16;4\}}}}[/tex]
