If-R-are-the-roots-of-the-equation-ax-2-bx-c-0-k-R-lies-between-and-if-

Question Number 5206 by ´[ last updated on 30/Apr/16

If α, β ∈ R, are the roots of the equation ax^2 +bx+c=0, k ∈ R lies between α and β, if

$$\mathrm{If}\:\alpha,\:\beta\:\in\:{R},\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0},\:{k}\:\in\:{R}\:\mathrm{lies}\:\mathrm{between}\:\alpha\:\mathrm{and} \\ $$$$\beta,\:\mathrm{if} \\ $$

Commented by prakash jain last updated on 30/Apr/16

ax^2 +bx+c takes a sign opposite of a for α<k<β ⇒ak^2 +bk+c has the sign −a ⇒a^2 k^2 +abk+ac<0

$${ax}^{\mathrm{2}} +{bx}+{c}\:\mathrm{takes}\:\mathrm{a}\:\mathrm{sign}\:\mathrm{opposite}\:\mathrm{of}\:{a}\:\mathrm{for}\:\alpha<{k}<\beta \\ $$$$\Rightarrow{ak}^{\mathrm{2}} +{bk}+{c}\:\mathrm{has}\:\mathrm{the}\:\mathrm{sign}\:−{a} \\ $$$$\Rightarrow{a}^{\mathrm{2}} {k}^{\mathrm{2}} +{abk}+{ac}<\mathrm{0} \\ $$

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