Question

Aflaţi valorile numerice ale expresiilor pentru

valorile specificate:

a)$\frac{4}{x+2}$ pentru x aparține{0,2}

b) $\frac{x+1}{x^{2} +1}$ pentru x apartine {-1, $\sqrt{2}$ , $\sqrt{3}$, -1 }

Answer 1

Salutare!

$\color{red}\boxed{\boxed{\bf Formule: (a+b)(a-b)=a^{2}-b^{2}~si~(a\pm b)^{2}=a^{2} \pm2ab+b^{2}}}$

$\it~~$

$\bf a)~ \dfrac{4}{x+2},~pentru ~x\in \{0,2\}$

$\it~~~$

$\bf Daca ~\underline{x = 0} \implies \dfrac{4}{0+2}=\dfrac{4}{2}=\dfrac{\not4}{\not2}=\boxed{\boxed{\bf 2}}$

$\it~~~$    

$\bf Daca ~x = 2 \implies \dfrac{4}{2+2}=\dfrac{4}{4}=\boxed{\boxed{\bf 1}}$

$\it~~~$  

$\bf b)~ \dfrac{x+1}{x^{2}+1},~pentru ~x\in \{-1, \sqrt{2},\sqrt{3}-1\}$

$\it~~~$

$\bf Daca ~\underline{x = -1 }\implies \dfrac{-1+1}{(-1)^{2}+1}= \dfrac{0}{1+1}=\boxed{\boxed{\bf \dfrac{0}{2}}}$

$\it~~$

$\bf Daca~ \underline{x = \sqrt{2}} \implies \dfrac{\sqrt{2} +1}{(\sqrt{2} )^{2}+1}= \dfrac{\sqrt{2} +1}{2+1}=\boxed{\boxed{\bf \dfrac{\sqrt{2} +1}{3}}}$

$\it~~$

$\bf Daca~x = \sqrt{3}-1 \implies \dfrac{\sqrt{3}-1+1}{(\sqrt{3}-1)^{2}+1}=\dfrac{\sqrt{3}}{(\sqrt{3})^{2}-2\cdot\sqrt{3}+1+1}=\dfrac{\sqrt{3}}{3-2\sqrt{3}+2}$

$\it~~$

$\bf \dfrac{\sqrt{3}}{5-2\sqrt{3}}=\dfrac{\sqrt{3}}{5-2\sqrt{3}}\cdot\dfrac{5+2\sqrt{3}}{5+2\sqrt{3}}=\dfrac{\sqrt{3} (5+2\sqrt{3})}{(5-2\sqrt{3})(5+2\sqrt{3})}=$

$\it~~$

$\bf \dfrac{5\sqrt{3} +6}{5^{2}-(2\sqrt{3})^{2}}=\dfrac{5\sqrt{3} +6}{25- 4\cdot3} =\dfrac{5\sqrt{3} +6}{25-12}=\boxed{\boxed{\bf\dfrac{5\sqrt{3} +6}{13}}}$

$\it~~$

PS: Daca esti pe telefon, te rog sa glisezi spre dreapta pentru a vedea rezolvarea completa

#copaceibrainly