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Question Number 20054 by Tinkutara last updated on 21/Aug/17

$$\mathrm{If}\alpha \mathrm{and}\beta (\alpha <\beta )\mathrm{are}\mathrm{the}\mathrm{roots}\mathrm{of}\mathrm{the}$$$$\mathrm{equation}{x}^2+bx+c=0,\mathrm{where}$$$$c<0<b,\mathrm{then}$$$$\left(1\right)0<\alpha <\beta$$$$\left(2\right)\alpha <0<\beta <\mid \alpha \mid$$$$\left(3\right)\alpha <\beta <0$$$$\left(4\right)\alpha <0<\mid \alpha \mid <\beta$$

Answered by ajfour last updated on 21/Aug/17

$$c<0<b,\alpha +\beta =-b,\alpha \beta =c$$$$\Rightarrow x=\frac{\alpha +\beta }{2}=\frac{-b}{2}<0\mathrm{\&}\alpha \beta =c<0$$$$\Rightarrow \alpha <0and\beta >0$$$$furthersince\frac{\alpha +\beta }{2}<0,\mid \alpha \mid >\beta$$$$socorrectoptionis\left(2\right).$$

Commented by Tinkutara last updated on 21/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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