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

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Discussion Overview

The discussion revolves around the absolute value equation |x^2 + 4x| = -12, exploring the existence of real or complex solutions. Participants examine the properties of absolute values and their implications for the equation.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions whether there are no real number solutions and suggests that the answer might be imaginary in advanced math contexts.
  • Another participant asserts that there is no solution, real or otherwise, and prompts consideration of the lowest value that |x^2 + 4x| can take.
  • A third participant explains the definition of absolute value for real numbers, emphasizing that it is never negative, and extends this definition to complex numbers, noting that their absolute value is also non-negative.
  • A later reply expresses gratitude for clarification, indicating confusion about the existence of complex solutions.

Areas of Agreement / Disagreement

Participants generally agree that there are no solutions to the equation, but there is some uncertainty regarding the mention of complex solutions, which one participant questions.

Contextual Notes

The discussion does not resolve the implications of absolute value in relation to complex numbers or the specifics of the lowest value of |x^2 + 4x|.

mathdad
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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?
 
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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.
 
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 $$\sqrt{a^2+ b^2}$$. That is also never negative.
 
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.
 

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