x²+4x²/|x|=0
To solve the equation \( x^2 + \frac{4x^2}{|x|} = 0 \), let's first rewrite the equation for clarity:
\[ x^2 + \frac{4x^2}{|x|} = 0 \]
Next, we can identify two cases based on the value of \( x \) since the absolute value function, \( |x| \), behaves differently for positive and negative values of \( x \).
Case 1: \( x > 0 \)
In this case, \( |x| = x \). Substituting this into the equation gives:
\[ x^2 + \frac{4x^2}{x} = 0 \]
This simplifies to:
\[ x^2 + 4x = 0 \]
Factoring out \( x \):
\[ x(x + 4) = 0 \]
This results in two potential solutions:
- \( x = 0 \)
- \( x + 4 = 0 \Rightarrow x = -4 \) (but this does not apply since \( x > 0 \))
Thus, no solutions come from Case 1.
Case 2: \( x < 0 \)
In this case, \( |x| = -x \). Substituting this into the equation gives:
\[ x^2 + \frac{4x^2}{-x} = 0 \]
This simplifies to:
\[ x^2 - 4x = 0 \]
Factoring out \( x \):
\[ x(x - 4) = 0 \]
This results in two potential solutions:
- \( x = 0 \)
- \( x - 4 = 0 \Rightarrow x = 4 \) (but this does not apply since \( x < 0 \))
Thus, again, no solutions come from Case 2.
Conclusion
Since neither case yields a valid solution other than \( x = 0 \), we can conclude that the only solution to the equation \( x^2 + \frac{4x^2}{|x|} = 0 \) is:
\[ \boxed{0} \]
