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Question Number 215167 by MathematicalUser2357 last updated on 30/Dec/24

$$\mathrm{If}\mathrm{the}\mathrm{quadratic}\mathrm{equation}{x}^2-2x-m+1=0\mathrm{has}\mathrm{two}\mathrm{different}\mathrm{positive}\mathrm{real}\mathrm{roots}\alpha ,\beta ,$$$$\mathrm{Determine}\mathrm{the}\mathrm{range}\mathrm{of}m(\mathrm{For}\mathrm{example},0<m<3)\mathrm{or}\mathrm{I}\mathrm{will}\mathrm{force}\mathrm{you}\mathrm{to}\mathrm{determine}$$

Answered by A5T last updated on 30/Dec/24

$${\mathrm{x}}_{1,2}=\frac{2\underset{-}{+}\sqrt{4-4(1-\mathrm{m})}}{2}=1\underset{-}{+}\sqrt{\mathrm{m}}$$$$\mathrm{m}>0\wedge 1-\sqrt{\mathrm{m}}>0\Rightarrow \mathrm{m}>0\wedge \mathrm{m}<1$$$$\Rightarrow 0<\mathrm{m}<1$$

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