Solve the absolute value equation?

|x| = x^2 + x - 3

Assuming x >= 0, we have:
x = x^2 + x - 3
x^2 + x - x - 3 = 0
x^2 - 3 = 0
x^2 = 3
x = sqrt(3) or x = -sqrt(3) [Rejected, since > 0]

Assuming x < 0. We have:
-x = x^2 + x - 3
x^2 + 2x - 3 = 0
(x + 3)(x - 1) = 0
x + 3 = 0 or x - 1 = 0
x = -3 or x = 1 (rejected, since x < 0)

Hence, solutions are x = sqrt(3) or x = -3

|x| = x² + x - 3
x > 0, x² - 3 = 0 => x = √3
x < 0, x² + 2x - 3 = 0 => x = -3
x = √3 or x = -3

When solving absolute value problems, the equation must be solved twice: once assuming the content of the abs. value function is positive. After solving the equation, the solution must agree with the assumed sign. The same is then repeated for the content being negative.

For example, in the present case,
|x|=x^2+x-3
must be solve as follows:
1. x>0
x=x^2+x-3 => x^2-3=0 => x=+/-sqrt(3)
But -sqrt(3) must be rejected because the sign does not agree with our previous assumption that x>0.
So we retain x=sqrt(3).
2. x<0
-x=x^2+x-3 => x^2+2x-3=0 => (x+3)(x-1)=0
=> x=-3 or x=1
Again, we reject x=1 because it does not agree with our previous assumption that x<0.
So we retain x=-3.
In all,the solution gives x=-3 or x=sqrt(3).
Check and make sure the answers agree with the original equation.
(and the answer is not 2sqrt(2). )

if x≥0 then |x| = x
if x<0 then |x| = -x
This is the definition of absolute value.

|x|= x^2 + x - 3

x≥0
x=x²+x-3
x²-3=0
x=±√3 reject - as x≥0
x=√3

x<0
-x= x^2 + x - 3
x²+2x-3=0
(x+3)(x-1)=0
x=-3 or 1 reject as x<0

x=-3 or √3