I would like to show that the largest root of a second-degree polynomial, i.e., $a_2 x^2+a_1 x +a_0$ is greater than the largest root of another, $b_2 x^2+b_1 x +b_0$.
Is there a way to show this without computing the roots themselves?
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What conditions are given on the coefficients of the second degree polynomials? – Zain Patel Jul 08 '16 at 08:56
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$a_2$ and $b_2$ smaller than zero. – Marius Zoican Jul 08 '16 at 09:01
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Surely you'd need some relationship between the $a_i$ and the $b_i$, otherwise it's just arbitrary. – Zain Patel Jul 08 '16 at 10:11
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I was mostly asking for techniques/tricks. But, my real problem is: $f\left(\gamma\right)=-2 \alpha ^2 \delta (\gamma -1) (\gamma (\delta \mu +n-1)-\delta \mu n+n-1)-2 \alpha \delta \mu n (\delta \mu -2)+\mu n \left(4-\delta ^2 \mu ^2\right)$ and $g\left(\gamma\right)=-2 \alpha ^2 \delta (\gamma -1) (-\delta \mu +\gamma +1)-2 \alpha \delta \mu (\delta \mu -2)-\delta ^2 \mu ^3+4 \mu$ with $\alpha\in\left(0,1\right)$, $\delta\in\left(0,1\right)$, $\mu\in\left(0,1\right)$, $n\geq 2$. From manipulating the roots, the largest root of $f$ > the largest root of $g$. – Marius Zoican Jul 08 '16 at 11:21