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

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SUMMARY

The absolute value equation |x^2 + 4x| = -12 has no solutions in the set of real numbers. The absolute value function, defined as |x|, is always non-negative, meaning it cannot equal a negative number such as -12. The lowest value of |x^2 + 4x| occurs at its vertex, which is also non-negative. Therefore, the equation has no solutions, neither real nor complex, as the absolute value cannot yield a negative result.

PREREQUISITES
  • Understanding of absolute value functions
  • Familiarity with quadratic equations
  • Basic knowledge of complex numbers
  • Concept of function minima and maxima
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  • Study the properties of absolute value functions
  • Learn about the vertex form of quadratic equations
  • Explore complex number representations and their magnitudes
  • Investigate the implications of negative values in mathematical equations
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Mathematics students, educators, and anyone interested in understanding the limitations of real numbers in solving absolute value equations.

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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