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Resposta:

[tex]\textsf{Leia abaixo}[/tex]

Explicação passo a passo:

[tex]\sf \dfrac{x + \sqrt{3}}{\sqrt{x} + \sqrt{x + \sqrt{3}}} + \dfrac{x - \sqrt{3}}{\sqrt{x} - \sqrt{x - \sqrt{3}}} = \sqrt{x}[/tex]

[tex]\sf \dfrac{x + \sqrt{3}}{\sqrt{x} + \sqrt{x + \sqrt{3}}}\:.\:\dfrac{\sqrt{x} - \sqrt{x + \sqrt{3}}}{\sqrt{x} - \sqrt{x + \sqrt{3}}} + \dfrac{x - \sqrt{3}}{\sqrt{x} - \sqrt{x - \sqrt{3}}}\:.\:\dfrac{\sqrt{x} + \sqrt{x - \sqrt{3}}}{\sqrt{x} + \sqrt{x - \sqrt{3}}} = \sqrt{x}[/tex]

[tex]\sf \dfrac{x\sqrt{x} - x\sqrt{x + \sqrt{3}} + \sqrt{x}\sqrt{3} -\sqrt{3}\sqrt{x + \sqrt{3}}}{x - x - \sqrt{3}} + \dfrac{x\sqrt{x} + x\sqrt{x - \sqrt{3}} - \sqrt{x}\sqrt{3} -\sqrt{3}\sqrt{x - \sqrt{3}}}{x - x + \sqrt{3}} = \sqrt{x}[/tex]

[tex]\sf \dfrac{\sqrt{x}(x + \sqrt{3}) - (\sqrt{x + \sqrt{3}})(x + \sqrt{3})}{-\sqrt{3}} + \dfrac{\sqrt{x}(x - \sqrt{3}) + (\sqrt{x - \sqrt{3}})(x - \sqrt{3})}{\sqrt{3}} = \sqrt{x}[/tex]

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

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

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

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

[tex]\sf (x + \sqrt{3})^3 + 2\:.\:\sqrt{(x + \sqrt{3})^3\:.\:(x - \sqrt{3})^3} + (x - \sqrt{3})^3 = 27x[/tex]

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

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

[tex]\sf (x + \sqrt{3})^3 + (x - \sqrt{3})^3= 2x^3 + 18x[/tex]

[tex]\sf 2x^3 + 18x + 2\:.\:\sqrt{(x + \sqrt{3})^3\:.\:(x - \sqrt{3})^3} = 27x[/tex]

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

[tex]\sf \left(2\:.\:\sqrt{(x + \sqrt{3})^3\:.\:(x - \sqrt{3})^3}\right)^2 = (-2x^3 + 9x)^2[/tex]

[tex]\sf 4(x^2 - 3)^3 = 4x^6 - 36x^4 + 81x^2[/tex]

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

[tex]\sf 4x^6 - 36x^4 + 108x^2 - 108 = 4x^6 - 36x^4 + 81x^2[/tex]

[tex]\sf 27x^2 = 108[/tex]

[tex]\sf x^2 = 4[/tex]

[tex]\sf x = \pm\:\sqrt{4}[/tex]

[tex]\sf x = \pm\:2[/tex]

[tex]\sf \sqrt{x} \rightarrow x \geq 0[/tex]

[tex]\sf \sqrt{x + \sqrt{3}} \rightarrow x + \sqrt{3} \geq 0[/tex]

[tex]\sf \sqrt{x - \sqrt{3}} \rightarrow x - \sqrt{3} \geq 0[/tex]

[tex]\boxed{\boxed{\sf S = \{2\}}}[/tex]