how to solve the quadratic equation x^2-9|x|+20=0

First, you need to express the absolute value
|x|:

|x| = x, if
x>=0

|x| = -x, if x <
0

Therefore, you'll have to solve quadratic equation in
both cases.

We'll start with the first case,
x>=0.

x^2 - 9x + 20 =
0

We'll apply quadratic
formula:

x1 =
[9+sqrt(81-80)]/2

x1 =
(9+1)/2

x1 = 5

x2 =
(9-1)/2

x2 = 4

Since both
values of x are positive, they represents the soutions of
equation.

We'll solve the quadratic for the second case,
x< 0:

x^2 + 9x + 20 =
0

x1 = [-9+sqrt(81-80)]/2

x1 =
(-9+1)/2

x1 = -4

x2 =
(-9-1)/2

x2 = -5

Since both
values are negative, they are also solutions of the
equation.

Therefore, all real solutions of
the quadratic module equation are {-5 ; -4 ; 4 ;
5}.