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Let-a-b-c-R-a-0-such-that-a-and-4a-3b-2c-have-the-same-sign-Show-that-the-equation-ax-2-bx-c-0-can-not-have-both-roots-in-the-interval-1-2-




Question Number 18384 by Tinkutara last updated on 19/Jul/17
Let a, b, c ∈ R, a ≠ 0, such that a and  4a + 3b + 2c have the same sign. Show  that the equation ax^2  + bx + c = 0 can  not have both roots in the interval  (1, 2).
$$\mathrm{Let}\:{a},\:{b},\:{c}\:\in\:{R},\:{a}\:\neq\:\mathrm{0},\:\mathrm{such}\:\mathrm{that}\:{a}\:\mathrm{and} \\ $$$$\mathrm{4}{a}\:+\:\mathrm{3}{b}\:+\:\mathrm{2}{c}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{sign}.\:\mathrm{Show} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{can} \\ $$$$\mathrm{not}\:\mathrm{have}\:\mathrm{both}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left(\mathrm{1},\:\mathrm{2}\right). \\ $$
Answered by Tinkutara last updated on 22/Jul/17
Let α and β be the roots of the equation.  Let a ≥ 0 so that 4a + 3b + 2c ≥ 0  ⇒ 4 + 3(b/a) + 2(c/a) ≥ 0  ⇒ 4 − 3(α + β) + 2αβ ≥ 0  ⇒ (α − 1)(β − 2) + (α − 2)(β − 1) ≥ 0,  which is not possible if both α and β  belong to (1, 2).
$$\mathrm{Let}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{be}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}. \\ $$$$\mathrm{Let}\:{a}\:\geqslant\:\mathrm{0}\:{so}\:{that}\:\mathrm{4}{a}\:+\:\mathrm{3}{b}\:+\:\mathrm{2}{c}\:\geqslant\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{4}\:+\:\mathrm{3}\frac{{b}}{{a}}\:+\:\mathrm{2}\frac{{c}}{{a}}\:\geqslant\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{4}\:−\:\mathrm{3}\left(\alpha\:+\:\beta\right)\:+\:\mathrm{2}\alpha\beta\:\geqslant\:\mathrm{0} \\ $$$$\Rightarrow\:\left(\alpha\:−\:\mathrm{1}\right)\left(\beta\:−\:\mathrm{2}\right)\:+\:\left(\alpha\:−\:\mathrm{2}\right)\left(\beta\:−\:\mathrm{1}\right)\:\geqslant\:\mathrm{0}, \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{not}\:\mathrm{possible}\:\mathrm{if}\:\mathrm{both}\:\alpha\:\mathrm{and}\:\beta \\ $$$$\mathrm{belong}\:\mathrm{to}\:\left(\mathrm{1},\:\mathrm{2}\right). \\ $$

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