Question

Aflati x astfel incit radical din 15x pe 6 apartine multimii Z

Answer 1

$\sqrt{\dfrac{15x}{6}} =\sqrt{\dfrac{3\cdot5x}{3\cdot2}}=\sqrt{\dfrac{5x}{2}}$

$\sqrt{\dfrac{5x}{2}} \in \mathbb{Z} \ \Longrightarrow \dfrac{5x}{2} = k^2 \ \Longrightarrow 5x=2k^2 \Longrightarrow x= \dfrac{2k^2}{5}\ \ \ (*)$,  k este din Z.

Din relatia  (*), in ipoteza ca x este numar intreg,  rezulta  ca :

$x\in \{0, 10, 40, 90, 160, 250, ...\},\ \ \ adica:\ \ x = 10 n^2,\ unde\ n \in \mathbb{Z}$

Verificarea este imediata : $\sqrt{\dfrac{15x}{6}} \ \ devine\ pentru\ x =10n^2$



$\sqrt{\dfrac{15\cdot10n^2}{6}} = \sqrt{\dfrac{150n^2}{6}} = \sqrt{25n^2} =5|n| \in \mathbb{Z}$

Answer 2

radical 15x/6 in Z.simplu:x va fi +radical 540 si -radical 540