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

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Simplifying x2 + -9x + 20 = 0

x2 -4x - 5 + 20

factor and group:

(x+-4)(x+-5) = 0

x = 4 x= 5

Simplifying x2 + -9x + 20 = 0

(x+-4)(x+-5) = 0

(x+-4)

x+4=0

x=-4

x-4=0

x=4

(x+-5)

x+5=0

x=-5

x-5=0

x=5

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}.**