Question

Rezolvati, in multimea numerelor reale, ecuatiile:

a) radical 3 * x + 3 = 0
b) -radical5 * x +10 = radical20 *x -20
c) y-radical2/3 + y+radical2/2=4radical2 + y
d) (y-radical3) * (y-radical5) = 0
e) /7 -radical7 * x / = 14

Answer 1

a)

$\sqrt{3} x + 3 = 0 \iff \sqrt{3} x = - 3$

$x = -\dfrac{3}{\sqrt{3} } \iff x = -\dfrac{3\sqrt{3}}{3}$

$\implies x = - \sqrt{3}$

b)

$-\sqrt{5} x + 10 = \sqrt{20} x -20$

$2\sqrt{5} x + \sqrt{5} x = 20 + 10$

$3\sqrt{5} x = 30 \iff x = \dfrac{30}{3\sqrt{5} }$

$x = \dfrac{30\sqrt{5}}{15} \implies x = 2\sqrt{5}$

c)

$\dfrac{y-\sqrt{2} }{3} + \dfrac{y+\sqrt{2} }{2} = 4\sqrt{2} + y \ \ \Big|\cdot 6$

$2(y - \sqrt{2}) + 3(y + \sqrt{2}) = 24\sqrt{2} + 6y$

$2y - 2\sqrt{2} + 3y + 3\sqrt{2} = 24\sqrt{2} + 6y$

$5y + \sqrt{2} = 24\sqrt{2} + 6y$

$6y - 5y = \sqrt{2} - 24\sqrt{2}$

$\implies y = -23\sqrt{2}$

d)

$(y - \sqrt{3}) \cdot (y - \sqrt{5}) = 0$

$y - \sqrt{3} = 0 \implies y = \sqrt{3}$

$y - \sqrt{5} = 0 \implies y = \sqrt{5}$

$\implies S = \Big\{\sqrt{3} ; \sqrt{5}\Big\}$

e)

$\Big|7 - \sqrt{7} x \Big| = 14 \iff 7 - \sqrt{7} x = \pm 14$

$7 - \sqrt{7} x = - 14 \iff \sqrt{7} x = 7 +14$

$\sqrt{7} x = 21 \iff x = \dfrac{21}{\sqrt{7} }$

$x = \dfrac{21\sqrt{7} }{7} \iff x = 3\sqrt{7}$

$7 - \sqrt{7} x = 14 \iff \sqrt{7} x = 7 - 14$

$\sqrt{7} x = -7 \iff x = -\dfrac{7}{\sqrt{7} }$

$x = -\dfrac{7\sqrt{7} }{7} \iff x = -\sqrt{7}$

$\implies S = \Big\{-\sqrt{7} ; 3\sqrt{7}\Big\}$