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Question Number 215166 by MathematicalUser2357 last updated on 30/Dec/24

$$\mathrm{If}\mathrm{the}\mathrm{quadratic}\mathrm{equation}3x^2+8x+2k=0\mathrm{has}\mathrm{two}\mathrm{different}\mathrm{negative}\mathrm{real}\mathrm{roots},$$$$\mathrm{Determine}\mathrm{the}\mathrm{range}\mathrm{of}k(\mathrm{For}\mathrm{example},0<k<3)\mathrm{or}\mathrm{I}\mathrm{will}\mathrm{force}\mathrm{you}\mathrm{to}\mathrm{determine}$$

Answered by A5T last updated on 30/Dec/24

$${\mathrm{x}}_{1,2}=\frac{-8\underset{-}{+}\sqrt{64-24\mathrm{k}}}{6}=\frac{-8\underset{-}{+}2\sqrt{16-6\mathrm{k}}}{6}$$$$=\frac{-4\underset{-}{+}\sqrt{16-6\mathrm{k}}}{3};$$$$16-6\mathrm{k}\ne 0\wedge 16-6\mathrm{k}>0\Rightarrow \mathrm{k}<\frac{8}{3}\wedge -4+\sqrt{16-6\mathrm{k}}<0$$$$\Rightarrow \mathrm{k}<\frac{8}{3}\wedge 16-6\mathrm{k}⩽16\Rightarrow \mathrm{k}>0$$$$\Rightarrow 0<\mathrm{k}<\frac{8}{3}$$

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