Question

Определите количество корней уравнения |х^2-4х|=а в зависимости от значения параметра а.

Answer 1

$|x^2-4x|=a, \ 1) a textless 0, xinvarnothing; \ 2) a=0, |x^2-4x|=0, \ x^2-4x=0, \ x(x-4)=0, \ x_1=0, x_2=4; \ 3) a textgreater 0, left [ {{x^2-4x=-a,} atop {x^2-4x=a;}} right. left [ {{x^2-4x+a=0,} atop {x^2-4x-a=0;}} right. \ left [ {{D_{/4}=(-2)^2-1cdot a,} atop {D_{/4}=(-2)^2-1cdot(-a);}} right. \ 3.1) left [ {{4-a textless 0,} atop {4+a textless 0;}} right. left [ {{a textgreater 4,} atop {a textless -4;}} right. \ a textgreater 4, xinvarnothing;$
$3.2) left [ {{4-a=0,} atop {4+a=0;}} right. left [ {{a=4,} atop {a=-4;}} right. a=4, \ x^2-4x+4=0, (x-2)^2=0, \ x_1=x_2=2; \ 3.3) left [ {{4-a textgreater 0,} atop {4+a textgreater 0;}} right. left [ {{a textless 4,} atop {a textgreater -4;}} right. 0 textless a textless 4, \ x_1=2-sqrt{4-a}, x_2=2+sqrt{4-a}. \ \ ain(-infty;0)cup(4;+infty) -   xinvarnothing, \ ain{0,4} - x_1=x_2, \ ain(0;4) - x_1 neq x_2.$