Question

помогите решить уровнения...​

Answer 1

Ответ:

$\boxed{ x = 6}$

Объяснение:

$\sqrt{x + 3} + \sqrt{3x - 9} = 6$

$\sqrt{x + 3} + \sqrt{3(x - 3)} = 6$

$\sqrt{x + 3} +\sqrt{3} \cdot \sqrt{(x - 3)} = 6$

ОДЗ:

$\displaystyle \left \{ {{x + 3\geq 0} \atop {x - 3\geq 0}} \right.$ $\displaystyle \left \{ {{x \geq - 3} \atop {x -\geq 3}} \right \Longrightarrow x \in [3;+\infty)$

$\sqrt{x + 3} < 6$

$(\sqrt{x + 3})^{2} < 6^{2}$

$x + 3< 36$

$x < 33$

ОДЗ: $x \in (3;33)$

$(\sqrt{x + 3} + \sqrt{3x - 9})^{2} = 6^{2}$

$x + 3 + 3x - 9 + 2\sqrt{(x + 3)(3x - 9)} = 36$

$4x - 6 + 2\sqrt{3(x + 3)(x - 3)} = 36$

$4x - 6 + 2\sqrt{3(x + 3)(x - 3)} = 36$

$2\sqrt{3(x^{2} - 9)} = 42 - 4x|:2$

$\sqrt{3(x^{2} - 9)} = 21 - 2x$

$(\sqrt{3x^{2} - 27})^{2} = (21 - 2x)^{2}$

$3x^{2} - 27 = 441 - 84x + 4x^{2}$

$x^{2} - 84x + 468 = 0 |+1296$

$x^{2} - 84x + 468 + 1296 = 1296$

$x^{2} - 84x + 1764 = 1296$

$(x - 42)^{2} = 36^{2}$

$\sqrt{ (x - 42)^{2} }= \sqrt{ 36^{2}}$

$|x - 42| = 36$

$x - 42 = 36; x - 42 =-36$

$x_{1} = 78;x_{2} = 6$

$x_{1} \notin (3;33)$

$x = 6$