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

What is the sum of all the solutions of the equation ∣2x+8∣^2 −∣9x+36∣−9=0

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

Answered by bemath last updated on 10/Sep/20

the equation equivalent to 4∣x+4∣^2 −9∣x+4∣−9 = 0 let ∣x+4∣ = λ →4λ^2 −9λ−9=0 →(4λ+3)(λ−3)=0 λ = −(3/4)(rejected), for λ= 3 (acceptable) →∣x+4∣=3→ { ((x_1 +4=−3)),((x_2 +4= 3)) :} ⇔ x_1 +x_2 = −8

$$\mathrm{the}\:\mathrm{equation}\:\mathrm{equivalent}\:\mathrm{to}\: \\ $$$$\mathrm{4}\mid\mathrm{x}+\mathrm{4}\mid^{\mathrm{2}} −\mathrm{9}\mid\mathrm{x}+\mathrm{4}\mid−\mathrm{9}\:=\:\mathrm{0} \\ $$$$\mathrm{let}\:\mid\mathrm{x}+\mathrm{4}\mid\:=\:\lambda\: \\ $$$$\rightarrow\mathrm{4}\lambda^{\mathrm{2}} −\mathrm{9}\lambda−\mathrm{9}=\mathrm{0} \\ $$$$\rightarrow\left(\mathrm{4}\lambda+\mathrm{3}\right)\left(\lambda−\mathrm{3}\right)=\mathrm{0} \\ $$$$\lambda\:=\:−\frac{\mathrm{3}}{\mathrm{4}}\left(\mathrm{rejected}\right),\:\mathrm{for}\:\lambda=\:\mathrm{3}\:\left(\mathrm{acceptable}\right) \\ $$$$\rightarrow\mid\mathrm{x}+\mathrm{4}\mid=\mathrm{3}\rightarrow\begin{cases}{\mathrm{x}_{\mathrm{1}} +\mathrm{4}=−\mathrm{3}}\\{\mathrm{x}_{\mathrm{2}} +\mathrm{4}=\:\mathrm{3}}\end{cases} \\ $$$$\Leftrightarrow\:\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} \:=\:−\mathrm{8}\: \\ $$

Commented by MJS_new last updated on 10/Sep/20

funny: you get the right sum out of the wrong solution putting λ=x+4 ⇔ x=λ−4 we get 4λ^2 −9∣λ∣−9=0 { ((λ<0∧4λ^2 +9λ−9=0 ⇒ λ=−3)),((λ≥0∧4λ^2 −9λ−9=0 ⇒ λ=3)) :} ⇒ x_1 =−7∧x_2 =−1

$$\mathrm{funny}:\:\mathrm{you}\:\mathrm{get}\:\mathrm{the}\:\mathrm{right}\:\mathrm{sum}\:\mathrm{out}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{wrong}\:\mathrm{solution} \\ $$$$\mathrm{putting}\:\lambda={x}+\mathrm{4}\:\Leftrightarrow\:{x}=\lambda−\mathrm{4}\:\mathrm{we}\:\mathrm{get} \\ $$$$\mathrm{4}\lambda^{\mathrm{2}} −\mathrm{9}\mid\lambda\mid−\mathrm{9}=\mathrm{0} \\ $$$$\begin{cases}{\lambda<\mathrm{0}\wedge\mathrm{4}\lambda^{\mathrm{2}} +\mathrm{9}\lambda−\mathrm{9}=\mathrm{0}\:\Rightarrow\:\lambda=−\mathrm{3}}\\{\lambda\geqslant\mathrm{0}\wedge\mathrm{4}\lambda^{\mathrm{2}} −\mathrm{9}\lambda−\mathrm{9}=\mathrm{0}\:\Rightarrow\:\lambda=\mathrm{3}}\end{cases} \\ $$$$\Rightarrow\:{x}_{\mathrm{1}} =−\mathrm{7}\wedge{x}_{\mathrm{2}} =−\mathrm{1} \\ $$

Commented by bemath last updated on 10/Sep/20

wkwkskwk...typo sir. thank you

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

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