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Question Number 11834 by Mr Chheang Chantria last updated on 02/Apr/17

$$\mathit{P}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathrm{\forall }\mathit{x},\mathit{y}\in \mathit{R}$$ $$\Rightarrow 7{\mathit{x}}^2-6\mathit{x}\mathit{y}+2{\mathit{y}}^2+\mathit{x}+3>0$$

Answered by mrW1 last updated on 02/Apr/17

$$7{\mathit{x}}^2-6\mathit{x}\mathit{y}+2{\mathit{y}}^2+\mathit{x}+3$$ $$={\left(\frac{3}{\sqrt{2}}x\right)}^2-2\times \frac{3}{\sqrt{2}}x\times \sqrt{2}y+{\left(\sqrt{2}y\right)}^2+{\left(\frac{1}{2}x\right)}^2+2\times \frac{1}{2}x+1+[7-{\left(\frac{3}{\sqrt{2}}\right)}^2-{\left(\frac{1}{2}\right)}^2]x^2+(3-1)$$ $$={(\frac{3}{\sqrt{2}}x-\sqrt{2}y)}^2+{(\frac{1}{2}x+1)}^2+{\left(\frac{3}{2}x\right)}^2+2$$ $$>2>0$$

Commented byMr Chheang Chantria last updated on 03/Apr/17

$$\mathit{G}\mathit{o}\mathit{o}\mathit{d}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n};)$$ $$$$

Answered by ajfour last updated on 02/Apr/17

$$6x^2-6xy+2y^2+x^2+x+3$$ $$=2x^2[3-\frac{3y}{x}+{\left(\frac{y}{x}\right)}^2]+{(x+\frac{1}{2})}^2-\frac{1}{4}+3$$ $$=2x^2[{(\frac{y}{x}-\frac{3}{2})}^2-\frac{9}{4}+3]+{(x+\frac{1}{2})}^2+\frac{11}{4}$$ $$=2x^2[\frac{{(2y-3x)}^2}{4x^2}+\frac{3}{4}]+{(x+\frac{1}{2})}^2+\frac{11}{4}$$ $$=\frac{{(2y-3x)}^2}{2}+\frac{3x^2}{2}+{(x+\frac{1}{2})}^2+\frac{11}{4}>0$$

Commented byMr Chheang Chantria last updated on 03/Apr/17

$${\mathbf{that}}^{\prime }\mathbf{s}\mathbf{so}\mathbf{sweet};)$$ $$$$

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 02/Apr/17

$$7x^2+(1-6y)x+2y^2+3=0$$ $$\mathrm{\Delta }={(1-6y)}^2-4\times 7(2y^2+3)=$$ $$1-12y+36y^2-56y^2-84=$$ $$-20y^2-12y-83=-(20y^2+12y+83)$$ $${\mathrm{\Delta }}^{{}^{\prime }}={12}^2-4\times 20\times 83<0$$ $$because{\mathrm{\Delta }}^{\prime }<0,then\mathrm{\Delta }onlyhaveone$$ $$signthatsimilartothecoefficentof$$ $$y^{2}i.e:(-20).soalwyes\mathrm{\Delta }<0andthe$$ $$7x^2-6xy+2y^2+3,anywerehaveone$$ $$signthatsimilartosignof{x}^2,i.e:(+7)$$ $$sothispolynomialispositiveanywere.$$

Commented bymrW1 last updated on 02/Apr/17

$$goodandinterestingpointofview!$$

Commented byMr Chheang Chantria last updated on 03/Apr/17

$$\mathit{n}\mathit{i}\mathit{c}\mathit{e}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{a}\mathit{n}\mathit{d}\mathit{n}\mathit{i}\mathit{c}\mathit{e}\mathit{e}\mathit{x}\mathit{p}\mathit{l}\mathit{a}\mathit{i}\mathit{n}$$ $$\mathit{T}\mathit{h}\mathit{a}\mathit{n}\mathit{k}\mathit{s}\mathit{y}\mathit{o}\mathit{u};)$$

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