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Question Number 60734 by Tawa1 last updated on 25/May/19

$$\mathrm{Find}\mathrm{the}\mathrm{product}\mathrm{of}\mathrm{the}\mathrm{real}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{equation}$$$${(\mathrm{x}+2+\sqrt{{\mathrm{x}}^2+4\mathrm{x}+3})}^5-32{(\mathrm{x}+2-\sqrt{{\mathrm{x}}^2+4\mathrm{x}+3})}^5=31$$

Commented by Prithwish sen last updated on 25/May/19

$$\because (\mathrm{x}+2+\sqrt{{\mathrm{x}}^2+4\mathrm{x}+3})=\frac{1}{(\mathrm{x}+2-\sqrt{{\mathrm{x}}^2+4\mathrm{x}+3})}$$$$\mathrm{Now}\mathrm{let}$$$$(\mathrm{x}+2+\sqrt{{\mathrm{x}}^2+4\mathrm{x}+3})=\mathrm{a}$$$$\mathrm{then}\mathrm{the}\mathrm{equation}$$$${\mathrm{a}}^5-\frac{32}{{\mathrm{a}}^5}=31$$$$\Rightarrow ({\mathrm{a}}^5+1)-32\left(\frac{{\mathrm{a}}^5+1}{{\mathrm{a}}^5}\right)=0$$$$\mathrm{either}\mathrm{or},$$$${\mathrm{a}}^5=-1{\left(\frac{2}{\mathrm{a}}\right)}^5=1$$$$\mathrm{is}\mathrm{it}\mathrm{ok}?$$

Commented by Tawa1 last updated on 25/May/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by MJS last updated on 25/May/19

$$\mathrm{great}!$$$$\mathrm{my}\mathrm{thinking}\mathrm{is}\mathrm{too}\mathrm{complicated}\mathrm{sometimes}$$

Commented by Prithwish sen last updated on 25/May/19

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Answered by MJS last updated on 25/May/19

$${(a+b)}^5-32{(a-b)}^5=31$$$$T_1-32T_2=31\Rightarrow T_1=31+32T_2$$$$\mathrm{trying}\mathrm{some}\mathrm{easy}\mathrm{numbers}\mathrm{I}\mathrm{found}\mathrm{these}:$$$$(a+b)=-1\wedge (a-b)=-1$$$$x+2\pm \sqrt{x^2+4x+3}=-1$$$$\sqrt{x^2+4x+3}=\pm (x+3)$$$$\sqrt{(x+3)(x+1)}=\pm (x+3)$$$$(x+3)(x+1)={(x+3)}^2$$$$\Rightarrow x=-3\mathrm{in}\mathrm{both}\mathrm{cases}$$$$$$$${(a+b)}^5=32\wedge {(a-b)}^5=\frac{1}{32}$$$$x+2+\sqrt{x^2+4x+3}=\sqrt[5]{32}x+2-\sqrt{x^2+4x+3}=\frac{1}{\sqrt[5]{32}}$$$$\sqrt{x^2+4x+3}=-x\sqrt{x^2+4x+3}=x+\frac{3}{2}$$$$x^2+4x+3=x^2{x}^2+4x+3={(x+\frac{3}{2})}^2$$$$x=-\frac{3}{4}x=-\frac{3}{4}$$$$$$$$\mathrm{of}\mathrm{course}\mathrm{we}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{yet}\mathrm{know}\mathrm{if}\mathrm{there}\mathrm{are}\mathrm{more}$$$$\mathrm{solutions}$$

Commented by Tawa1 last updated on 25/May/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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