Question

Rezolvați ecuația 3x^2= (15-3√3)√28-10√3
28-10√3 e totul sub radical.

Answer 1

$3x^2= (15-3\sqrt 3)\sqrt{28-10\sqrt 3} \\ \\ \begin{array}{rcl} 28-10\sqrt 3 &=& 28 - 2\cdot 5\sqrt 3\\ &=& 25 - 2\cdot 5\sqrt 3 + 3 \\ &=& (5-\sqrt{3})^2 \end{array} \\ \\ 3x^2 = (15-3\sqrt3)\sqrt{(5-\sqrt3)^2} \\ 3x^2 = (15-3\sqrt3)\big|\underset{\ \textgreater \ 0}{\underbrace{5-\sqrt 3}} \big| \\ 3x^2 = (15-3\sqrt 3)(5-\sqrt3)$

$3x^2 = 3(5-\sqrt 3)(5-\sqrt 3)\\ 3x^2 = 3(5-\sqrt3)^2\Big|:3 \\ x^2 = (5-\sqrt3)^2 \\ \\ \Rightarrow x_1 = (5-\sqrt3);\quad x_2 = -(5-\sqrt 3) \Rightarrow \\ \\ \Rightarrow \boxed{S = \Big\{5-\sqrt 3;~\sqrt3 -5\Big\}}$