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Question Number 112838 by bobhans last updated on 10/Sep/20

$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{all}\mathrm{the}\mathrm{solutions}$$$$\mathrm{of}\mathrm{the}\mathrm{equation}\mid 2\mathrm{x}+8{\mid }^2-\mid 9\mathrm{x}+36\mid -9=0$$

Answered by bemath last updated on 10/Sep/20

$$\mathrm{the}\mathrm{equation}\mathrm{equivalent}\mathrm{to}$$$$4\mid \mathrm{x}+4{\mid }^2-9\mid \mathrm{x}+4\mid -9=0$$$$\mathrm{let}\mid \mathrm{x}+4\mid =\lambda$$$$\to 4{\lambda }^2-9\lambda -9=0$$$$\to (4\lambda +3)(\lambda -3)=0$$$$\lambda =-\frac{3}{4}\left(\mathrm{rejected}\right),\mathrm{for}\lambda =3\left(\mathrm{acceptable}\right)$$$$\to \mid \mathrm{x}+4\mid =3\to \{\begin{array}{l}{\mathrm{x}}_1+4=-3\\ {\mathrm{x}}_2+4=3\end{array}$$$$\iff {\mathrm{x}}_1+{\mathrm{x}}_2=-8$$

Commented by MJS_new last updated on 10/Sep/20

$$\mathrm{funny}:\mathrm{you}\mathrm{get}\mathrm{the}\mathrm{right}\mathrm{sum}\mathrm{out}\mathrm{of}\mathrm{the}$$$$\mathrm{wrong}\mathrm{solution}$$$$\mathrm{putting}\lambda =x+4\iff x=\lambda -4\mathrm{we}\mathrm{get}$$$$4{\lambda }^2-9\mid \lambda \mid -9=0$$$$\{\begin{array}{l}\lambda <0\wedge 4{\lambda }^2+9\lambda -9=0\Rightarrow \lambda =-3\\ \lambda ⩾0\wedge 4{\lambda }^2-9\lambda -9=0\Rightarrow \lambda =3\end{array}$$$$\Rightarrow x_1=-7\wedge x_2=-1$$

Commented by bemath last updated on 10/Sep/20

$$\mathrm{wkwkskwk}...\mathrm{typo}\mathrm{sir}.\mathrm{thank}\mathrm{you}$$

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