Ensino Médio ⇒ (Iezzi) Equação Irracional Tópico resolvido

[mahriana]

Resolva em [tex3]\mathbb{R}[/tex3] a equação :

[tex3]\sqrt{x}- \sqrt{x-\sqrt{1-x}}=1[/tex3]

Resposta

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S = [tex3]\left \{ \right \}[/tex3]

[jrneliodias]

Olá, Mahriana.

[tex3]\sqrt x-1=\sqrt{x-\sqrt{1-x}}\,\,\,\,\,\Rightarrow\,\,\,\,\left(\sqrt x-1\right)^2=\left(\sqrt{x-\sqrt{1-x}}\right)^2\,\,\,\,\,e\,\,\,\,\,\,\sqrt x-1\geq[/tex3] [tex3]0[/tex3]

[tex3]\,\,\,\,\,\Rightarrow\,\,\,\,x-2\sqrt x+1=x-\sqrt{1-x}\,\,\,\,\,\,\,e\,\,\,\,\boxed{x\geq1}[/tex3]

[tex3]\Rightarrow\,\,\,\,\sqrt{1-x}+1=2\sqrt{x}\,\,\,\,\,\Rightarrow\,\,\,\,1-x+2\sqrt{1-x}+1=4x\\\\\,\,\,\,\,\Rightarrow\,\,\,\,2\sqrt{1-x}=5x-2\,\,\,\,\,\,\,e\,\,\,\,5x-2\geq0[/tex3]

[tex3]\,\,\,\,\,\Rightarrow\,\,\,\,4-4x=25x^2-20x+4\,\,\,\,e\,\,\,\,\,x\geq\frac{2}{5}[/tex3]

[tex3]\,\,\,\,\,\Rightarrow\,\,\,\,25x^2+16x=0\,\,\,\,e\,\,\,\,\,x\geq\frac{2}{5}[/tex3]

[tex3]\Rightarrow\,\,\,\,\,\,\,x=0\,\,\,\,\,ou\,\,\,\,\,x=\frac{-16}{25}\,\,\,\,\,\,\,\,e\,\,\,\,\,x\geq\frac{2}{5}[/tex3]

Fazendo a interseção, teremos:

[tex3]\text{S}=\{\}[/tex3]

Abraço, espero ter ajudado