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Not so standard inequality involving absolute values, roots

Source: Germany 2006 Grade 11-13 - #5

June 5, 2015
inequalitiesabsolute valuepolynomialRoot

Problem Statement

Let x≠0x \neq 0 be a real number satisfying ax2+bx+c=0ax^2+bx+c=0 with a,b,c∈Za,b,c \in \mathbb{Z} obeying ∣a∣+∣b∣+∣c∣>1|a|+|b|+|c| > 1. Then prove ∣x∣≥1∣a∣+∣b∣+∣c∣−1. |x| \geq \frac{1}{|a|+|b|+|c|-1}.
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