[tex]E(x) = \frac{x}{ \sqrt{2} } - (2 - \sqrt{2} ) + |3 - \sqrt{8} | \\ \\ E(x) = \frac{x \sqrt{2} }{2} - 2 + \sqrt{2} + 3 - 2 \sqrt{2} \\ \\ E(x) = \frac{x \sqrt{2} }{2} + {}^{ {}^{2)} } 1 - {}^{ {}^{2)} } \sqrt{2} \\ \\ E(x) = \frac{x \sqrt{2} + 2 - 2 \sqrt{2} }{2} \\ \\ \implies \text{ca \: sa \: fie \: natural \: trebuie \: sa} \\ \text{scapam \: de \: radicali \: prin} \\ \text{reducere(trebuie \: sa \: fie \: la \: fel)} \\ \implies x \sqrt{2} = 2 \sqrt{2} \\\implies\boxed{ x = 2}[/tex]

Explicație pas cu pas:

[tex]3 - \sqrt{8} = \sqrt{9} - \sqrt{8} > 0 \\ \implies |3 - \sqrt{8} | = 3 - 2 \sqrt{2} [/tex]

[tex]E(x) = (x \div \sqrt{2}) - (2 - \sqrt{2} ) + |3 - \sqrt{8} | = \\ = \dfrac{x}{ \sqrt{2} } - 2 + \sqrt{2} + 3 - 2 \sqrt{2} = \dfrac{x \sqrt{2} }{2} - \sqrt{2} + 1 \\ = \dfrac{x \sqrt{2} - 2 \sqrt{2} }{2} + 1 = \dfrac{(x - 2) \sqrt{2} }{2} + 1 \in \mathbb{N}[/tex]

[tex]\dfrac{(x - 2) \sqrt{2} }{2} \in \mathbb{N} \iff x - 2 = 0 \\ \bf \implies x = 2[/tex]