Question

Considerăm numerele u=\sqrt{9+4\sqrt{5} } si v=\sqrt{9-4\sqrt{5} } Dacă numerele raţionale a şi b verifică egalitatea au+bv=2020, atunci valoarea sumei 2a+b este egală cu

Si am 5 variante de raspuns:
505\sqrt{5}
505
-505
1010
-1010

DAU COROANA[tex][/tex]

Answer 1

$\it 9+4\sqrt5=5+4+4\sqrt5 =(\sqrt5)^2 +4\sqrt5+2^2=(\sqrt5+2)^2\\ \\ Analog,\ 9-4\sqrt5=(\sqrt5-2)^2\\ \\ u=\sqrt{(\sqrt5+2)^2} =\sqrt5+2\\ \\ v=\sqrt{(\sqrt5-2)^2} =|\sqrt5-2|=\sqrt5-2$

$\it au+bv =2020 \Rightarrow a(\sqrt5+2)+b(\sqrt5-2)=2020 \Rightarrow \\ \\ a\sqrt5+2a+b\sqrt5-2b=2020\Rightarrow \sqrt5(a+b) +2(a-b)=2020\Rightarrow \\ \\ \Rightarrow \begin{cases} \it a+b=0 \Rightarrow b=-a\ \ \ \ \ (1)\\ \\ \it 2(a-b)=2020|_{:2} \Rightarrow a-b=1010\stackrel{(1)}{\Longrightarrow} 2a=1010 \Rightarrow a=505\ \ \ (2)\end{cases}\\ \\ \\ (1),\ (2) \Rightarrow b=-505$

$\it 2a+b=2\cdot505-505=505$