MHB Challenge for Polynomial with Complex Coefficients

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The discussion centers on proving that the roots of the quadratic polynomial $ax^2 + bx + c$, where $a$ and $b$ are non-zero complex coefficients, lie within a specific region defined by the inequality $|x| \le \left|\dfrac{b}{a}\right| + \left|\dfrac{c}{b}\right|$. Participants explore various approaches to demonstrate this result, emphasizing the importance of understanding the behavior of complex numbers in polynomial equations. The hint provided suggests a potential method for tackling the proof, though specific techniques are not detailed in the summary. The focus remains on establishing the boundary conditions for the roots based on the coefficients. Ultimately, the discussion aims to solidify the understanding of polynomial roots in the context of complex coefficients.
anemone
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Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region

$|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.
 
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Hint:

Consider $|\sqrt{b^2-4ac}|$.
 

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