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Question Number 79085 by key of knowledge last updated on 22/Jan/20

$$\alpha \mathrm{and}\beta \left(\mathrm{or}\mathrm{more}\right)\mathrm{are}\mathrm{root}\mathrm{of}:$$ $${({\mathrm{x}}^2+\mathrm{x})}^2-2({\mathrm{x}}^2+\mathrm{x})-\mathrm{k}=0$$ $$\mathrm{and}\alpha \beta <0\Rightarrow \mathrm{k}\in ?$$

Commented bymr W last updated on 22/Jan/20

$$k\in [-1,0)$$ $$right?$$

Answered by MJS last updated on 23/Jan/20

$${(x^2+x)}^2-2(x^2+x)-k=0$$ $$\Rightarrow$$ $$x^2+x=1\pm \sqrt{k+1}$$ $$\Rightarrow$$ $$x=-\frac{1}{2}\pm \frac{\sqrt{5+4\sqrt{k+1}}}{2}\vee x=-\frac{1}{2}\pm \frac{\sqrt{5-4\sqrt{k+1}}}{2}$$ $$\mathrm{to}\mathrm{get}4\mathrm{real}\mathrm{solutions}$$ $$-1⩽k⩽\frac{9}{16}$$ $$\mathrm{for}k⩾-1{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{always}\mathrm{a}\mathrm{pair}\mathrm{of}\mathrm{roots}\alpha ,\beta$$ $$\mathrm{with}\alpha \beta <0$$

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