b) Expresia E(x) = x² + x + 1, pentru orice x = R \ {-2, -1, +√2}
Demonstrați că E(1-√2) < 1
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[tex]E(1-\sqrt{2}) = (1-\sqrt{2})^2 + (1-\sqrt{2}) + 1 =[/tex]
[tex]= \underline{1^2 - 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2} + 1 - \sqrt{2} + 1[/tex]
[tex]= 1 - 2\sqrt{2} + 2 - \sqrt{2} + 2[/tex]
[tex]= 5 - 3\sqrt{2} = 5 - \sqrt{18}[/tex]
[tex]\sqrt{18} > 4 \Rightarrow - \sqrt{18} < - 4 \Rightarrow - \sqrt{18} + 5 < - 4 + 5 \Rightarrow 5 - \sqrt{18} < 1[/tex]
[tex]\Rightarrow 5 - 3\sqrt{2} < 1 \Rightarrow \boldsymbol{ E(1-\sqrt{2} ) < 1}[/tex]
q.e.d.
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Formule de calcul prescurtat:
[tex]\boxed{\boxed{\boldsymbol{(a \pm b)^{2} = a^{2} \pm 2ab + b^{2}}} \ \ \boxed{\boldsymbol{(a - b)(a + b) = a^{2} - b^{2}}}}\\[/tex]
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