Prove an inequality involving a root of a quadratic equationIf x=ρis a solution to: x2+bx+c=0Prove that |ρ|−1&lt;|b|+|c|

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Prove an inequality involving a root of a quadratic equationIf x=ρis a solution to: x2+bx+c=0Prove that |ρ|−1&amp;lt;|b|+|c|

Tristatex9tw · Accepted Answer

Step 1Let ρ1 and ρ2 be the solutions to x2+bx+c=0/Let&#039;s assume, without loss of generality, that |ρ1≥|ρ2∣We shall consider two cases:a) If |ρ1|&amp;lt;1, then|ρ2|−1≤|ρ1|−1&amp;lt;0≤|b|+|c|b) If |ρ1|≥1, it follows from Vieta&#039;s formulas and the reverse triangular inequality that|b|+|c|=|ρ1+ρ2|+|ρ1ρ2|≥||ρ1|−|ρ2||+|ρ1||ρ2|=|ρ1|−|ρ2|+|ρ1||ρ2|=(|ρ1|−1)(|ρ2|+1)+1&amp;gt;(|ρ1|−1)(|ρ2|+1)≥|ρ1|−1≥|ρ2|−1.Therefore, in both cases it is true that |ρ|−1&amp;lt;|b|+|c|

microsgopx6z7 · Accepted Answer

Step 1Consider the matrixA=[01−c−b]then x2+bx+c is the corresponding characteristic polynomial.By Gershgorin&#039;s theorem, we have |ρ|≤1 or|ρ+b|≤|c|By reversed triangle inequality, |ρ|≤1 or |ρ|≤|c|+|b|,hence |ρ|≤max(1,|b|+|c|)≤1+|b|+|c|

Prove an inequality involving a root of a

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