Question

Se considera numerele reale a, b, x, y astfel încât
at 19 - 45 și
CERINTA
Atunci expresia ax - ay + bx - by este egală cu:
RASPUNSURI
Alege raspunsul corect:
1​

Answer 1

 

$\displaystyle\bf\\a+b=\sqrt{9-4\sqrt{5}}=\\\\=\sqrt{5+4-2\times\sqrt{5}\times2}=\\\\=\sqrt{5-2\times\sqrt{5}\times2+4}=\\\\=\sqrt{\left(\sqrt{5}\right)^2-2\times\sqrt{5}\times2+2^2}=\\\\=\sqrt{\left(\sqrt{5}-2\right)^2}=\sqrt{5}-2\\\\\boxed{\bf a+b=\sqrt{5}-2}$

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$\displaystyle\bf\\x-y=\sqrt{9+4\sqrt{5}}=\\\\=\sqrt{5+4+2\times\sqrt{5}\times2}=\\\\=\sqrt{5+2\times\sqrt{5}\times2+4}=\\\\=\sqrt{\left(\sqrt{5}\right)^2+2\times\sqrt{5}\times2+2^2}=\\\\=\sqrt{\left(\sqrt{5}+2\right)^2}=\sqrt{5}+2\\\\\boxed{\bf x-y=\sqrt{5}+2}\\\\\\ax-ay+bx-by=\\\\=ax+bx-ay-by=\\\\=x(a+b)-y(a+b)=\\\\=x(a+b)-y(a+b)=\\\\=(a+b)(x-y)=\\\\=\Big(\sqrt{5}-2\Big)\Big(\sqrt{5}+2\Big)=\\\\=\Big(\sqrt{5}\Big)^2-\Big(2\Big)^2=\\\\=5-4=\boxed{\bf1}\\\\Raspuns~corect~~\boxed{\bf B~~~1}$