IME / ITA ⇒ Operação com Radicais

[vinicius18]

Ache o valor de [tex3]x[/tex3]
[tex3]\frac{x+\sqrt{3}}{\sqrt{x}+\sqrt{x+\sqrt{3}}}+\frac{x - \sqrt{3}}{\sqrt{x}-\sqrt{x-\sqrt{3}}}=\sqrt{x}[/tex3]

[jedi]

[tex3]\frac{x+\sqrt{3}}{\sqrt{x}+\sqrt{x+\sqrt{3}}}+\frac{x - \sqrt{3}}{\sqrt{x}-\sqrt{x-\sqrt{3}}}=\sqrt{x}[/tex3]

[tex3]\frac{x+\sqrt{3}}{\sqrt{x}+\sqrt{x+\sqrt{3}}}.\left(\frac{\sqrt{x}-\sqrt{x+\sqrt{3}}}{\sqrt{x}-\sqrt{x+\sqrt{3}}}\right)+\frac{x - \sqrt{3}}{\sqrt{x}-\sqrt{x-\sqrt{3}}}.\left(\frac{\sqrt{x}+\sqrt{x-\sqrt{3}}}{\sqrt{x}+\sqrt{x-\sqrt{3}}}\right)=\sqrt{x}[/tex3]

[tex3]\frac{(x+\sqrt{3}).\left(\sqrt{x}-\sqrt{x+\sqrt{3}}\right)}{x-x-\sqrt{3}}+\frac{(x - \sqrt{3}).\left(\sqrt{x}+\sqrt{x-\sqrt{3}}\right)}{x-x+\sqrt{3}}=\sqrt{x}[/tex3]

[tex3]\frac{(x+\sqrt{3}).\left(\sqrt{x}-\sqrt{x+\sqrt{3}}\right)}{-\sqrt{3}}+\frac{(x - \sqrt{3}).\left(\sqrt{x}+\sqrt{x-\sqrt{3}}\right)}{\sqrt{3}}=\sqrt{x}[/tex3]

[tex3]-(x+\sqrt{3}).\left(\sqrt{x}-\sqrt{x+\sqrt{3}}\right)+(x - \sqrt{3}).\left(\sqrt{x}+\sqrt{x-\sqrt{3}}\right)=\sqrt{3}\sqrt{x}[/tex3]

[tex3]\cancel{-x\sqrt{x}}+x.\sqrt{x+\sqrt{3}}-\sqrt{3}.\sqrt{x}+\sqrt{3}.\sqrt{x+\sqrt{3}}+\cancel{x\sqrt{x}}+x\sqrt{x-\sqrt{3}} - \sqrt{3}\sqrt{x}-\sqrt{3}\sqrt{x-\sqrt{3}}=\sqrt{3}\sqrt{x}[/tex3]

[tex3]x.\sqrt{x+\sqrt{3}}+\sqrt{3}.\sqrt{x+\sqrt{3}}+x\sqrt{x-\sqrt{3}} -\sqrt{3}\sqrt{x-\sqrt{3}}-2\sqrt{3}.\sqrt{x}=\sqrt{3}\sqrt{x}[/tex3]

[tex3]x.\sqrt{x+\sqrt{3}}+\sqrt{3}.\sqrt{x+\sqrt{3}}+x\sqrt{x-\sqrt{3}} -\sqrt{3}\sqrt{x-\sqrt{3}}=\sqrt{3}\sqrt{x}+2\sqrt{3}.\sqrt{x}[/tex3]

[tex3]x.\sqrt{x+\sqrt{3}}+\sqrt{3}.\sqrt{x+\sqrt{3}}+x\sqrt{x-\sqrt{3}} -\sqrt{3}\sqrt{x-\sqrt{3}}=3\sqrt{3}\sqrt{x}[/tex3]

[tex3](x+\sqrt{3}).\sqrt{x+\sqrt{3}}+(x-\sqrt{3})\sqrt{x-\sqrt{3}}=3\sqrt{3}\sqrt{x}[/tex3]

[tex3]\sqrt{(x+\sqrt{3})^3}+\sqrt{(x-\sqrt{3})^3}=3\sqrt{3}\sqrt{x}[/tex3]

elevando os dois lados ao quadrado

[tex3](x+\sqrt3)^3+2\sqrt{(x+\sqrt{3})^3}.\sqrt{(x-\sqrt{3})^3}+(x-\sqrt3)^3=27x[/tex3]

[tex3](x+\sqrt3)^3+2\sqrt{(x+\sqrt{3})^3}.\sqrt{(x-\sqrt{3})^3}+(x-\sqrt3)^3=27x[/tex3]

[tex3]x^3+3\sqrt3.x^2+9x+3\sqrt3+2\sqrt{(x+\sqrt{3})^3}.\sqrt{(x-\sqrt{3})^3}+x^3-3\sqrt3.x^2+9x-3\sqrt3=27x[/tex3]

[tex3]2\sqrt{(x+\sqrt{3})^3}.\sqrt{(x-\sqrt{3})^3}=9x-2x^3[/tex3]

[tex3]2\sqrt{(x+\sqrt{3})^3(x-\sqrt{3})^3}=9x-2x^3[/tex3]

[tex3]2\sqrt{(x+\sqrt{3})^3(x-\sqrt{3})^3}=9x-2x^3[/tex3]

[tex3]2\sqrt{\left[(x+\sqrt{3})(x-\sqrt{3})\right]^3}=9x-2x^3[/tex3]

[tex3]2\sqrt{(x^2-3)^3}=9x-2x^3[/tex3]

elevando os dois lados ao quadrado

[tex3]4(x^2-3)^3=(9x-2x^3)^2[/tex3]

[tex3]4(x^6-9x^4+27x^2-27)=81x^2-36x^4+4x^6[/tex3]

[tex3]4x^6-36x^4+108x^2-108=81x^2-36x^4+4x^6[/tex3]

[tex3]27x^2-108=0[/tex3]

[tex3]x^2=4[/tex3]

[tex3]x=2[/tex3]