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

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The absolute value equation |x^2 + 4x| = -12 has no solutions in real numbers because the absolute value is always non-negative, meaning it cannot equal a negative number. The lowest value of |x^2 + 4x| occurs at its vertex, which is also non-negative. For complex numbers, the absolute value is defined as the square root of the sum of the squares of the real and imaginary parts, which is also non-negative. Therefore, there are no real or complex solutions to the equation. The discussion emphasizes the fundamental property of absolute values being non-negative.
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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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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