Предмет: Алгебра, автор: НоВыЙ13

Решите уравнение:
$2x^{2} + \sqrt{2 x^{2}-4x+12 } =4x+8$

Ответы

Автор ответа: dtnth

2

$2x^2+\sqrt{2x^2-4x+12}=4x+8$
ОДЗ: $2x^2-4x+12 \geq 0$
$x^2-2x+6 \geq 0$
$(x^2-2x+1)+5 \geq 0$
$(x-1)^2+5 \geq 0+5=5 \geq 0$

$2x^2-4x+12+\sqrt{2x^2-4x+12}-20=0$
замена
$\sqrt{2x^2-4x+12}=t \geq 0$
$t^2+t-20=0$
$(t+5)(t-4)=0$
$t+5=0;t_1=-5<0$ - не подходит
$t-4=0;t_2=4$
t=4
$\sqrt{2x^2-4x+12}=4$
$(\sqrt{2x^2-4x+12})^2=4^2$
$2x^2-4x+12=16$
$x^2-2x+6=8$
$x^2-2x+6-8=0$
$x^2-2x-2=0$
$D=(-2)^2-4*1*(-2)=4+8=12=4*3=2^2*3$
$x_{1,2}=\frac{2^+_-2\sqrt{3}}{2*1}=1^+_-\sqrt{3}$
ответ
$1^+_-\sqrt{3}$

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