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Question Number 18384 by Tinkutara last updated on 19/Jul/17

$$\mathrm{Let}a,b,c\in R,a\ne 0,\mathrm{such}\mathrm{that}a\mathrm{and}$$$$4a+3b+2c\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{sign}.\mathrm{Show}$$$$\mathrm{that}\mathrm{the}\mathrm{equation}{ax}^2+bx+c=0\mathrm{can}$$$$\mathrm{not}\mathrm{have}\mathrm{both}\mathrm{roots}\mathrm{in}\mathrm{the}\mathrm{interval}$$$$(1,2).$$

Answered by Tinkutara last updated on 22/Jul/17

$$\mathrm{Let}\alpha \mathrm{and}\beta \mathrm{be}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}.$$$$\mathrm{Let}a⩾0sothat4a+3b+2c⩾0$$$$\Rightarrow 4+3\frac{b}{a}+2\frac{c}{a}⩾0$$$$\Rightarrow 4-3(\alpha +\beta )+2\alpha \beta ⩾0$$$$\Rightarrow (\alpha -1)(\beta -2)+(\alpha -2)(\beta -1)⩾0,$$$$\mathrm{which}\mathrm{is}\mathrm{not}\mathrm{possible}\mathrm{if}\mathrm{both}\alpha \mathrm{and}\beta$$$$\mathrm{belong}\mathrm{to}(1,2).$$

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