Why can't real numbers satisfy this absolute value equation?

[mathdad]

Explain, in your own words, why there are no real numbers that satisfy the absolute value equation | x^2 + 4x | = - 12.

Can we say there is no real number solution here? If so, is the answer then imaginary taught in some advanced math class?

 

[Greg]

There's no solution, real or otherwise. What's the lowest value $|x^2+4x|$ can have? After answering that, consider the RHS of the equation.

 

[HOI]

For x a real number, |x| is defined as "x is x is non-negative, -x if x is negative". From that it should be clear that |x| is never negative.

For x a complex number, written as a+ bi, |x| is \(\displaystyle \sqrt{a^2+ b^2}\). That is also never negative.

 

[mathdad]

Thank you. Someone told me that there complex solutions but that did not make sense. So, I decided to ask the real math guys. Thank you again.