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Question Number 113857 by ayenisamuel last updated on 15/Sep/20

$$findthevaluesofkforwhichthe$$$$liney=kx-3doesnotmeetthecurve$$$$y=2x^2-3x+k.$$

Answered by MJS_new last updated on 15/Sep/20

$$\mathrm{first},\mathrm{intersect}\mathrm{them}$$$$kx-3=2x^2-3x+k$$$$x^2-\frac{k+3}{2}x+\frac{k+3}{2}=0$$$$\mathrm{let}x=t+\frac{k+3}{4}\mathrm{and}\mathrm{solve}\mathrm{for}{t}^2$$$$t^2=\frac{(k-5)(k+3)}{16}$$$$\Rightarrow \mathrm{if}\frac{(k-5)(k+3)}{16}<0\Rightarrow \mathrm{no}\mathrm{solution}$$$$\Rightarrow -3<k<5\mathrm{is}\mathrm{the}\mathrm{answer}$$

Answered by 1549442205PVT last updated on 16/Sep/20

$$\mathrm{We}\mathrm{see}\mathrm{that}$$$$liney=kx-3doesnotmeetthecurve$$$$y=2x^2-3x+k\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\mathrm{the}\mathrm{equation}$$$${2\mathrm{x}}^2-3\mathrm{x}+\mathrm{k}=\mathrm{kx}-3\mathrm{has}\mathrm{no}\mathrm{roots}$$$$\iff {2\mathrm{x}}^2-(3+\mathrm{k})\mathrm{x}+\mathrm{k}+3=0\mathrm{has}\mathrm{no}\mathrm{roots}$$$$\iff \mathrm{\Delta }={(3+\mathrm{k})}^2-4.2.(\mathrm{k}+3)<0$$$$\iff {\mathrm{k}}^2-2\mathrm{k}-15<0\iff (\mathrm{k}-5)(\mathrm{k}+3)<0$$$$\iff -3<\mathrm{k}<5$$$$\mathrm{Answer}\mathbf{is}:-3<\mathbf{k}<5$$

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