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Question Number 182073 by Acem last updated on 04/Dec/22

Find the sum of the solutions of the equation: ∣(√x) − 2∣+ (√x) ((√x) − 4)+ 2= 0 ; x> 0

$${Find}\:{the}\:{sum}\:{of}\:{the}\:{solutions}\:{of}\:{the}\:{equation}: \\ $$ $$\:\mid\sqrt{{x}}\:−\:\mathrm{2}\mid+\:\sqrt{{x}}\:\left(\sqrt{{x}}\:−\:\mathrm{4}\right)+\:\mathrm{2}=\:\mathrm{0}\:\:\:;\:{x}>\:\mathrm{0} \\ $$

Answered by HeferH last updated on 04/Dec/22

∣(√x) − 2∣ = 4(√x) − x − 2 (√x) − 2 = ± (4(√x) − x − 2) First eq. (√x)− 2 = −(4(√x) − x − 2) (√x) − 2 = x + 2 −4(√x) 5(√x) = x + 4 25x = x^2 + 16 + 8x x^2 − 17x + 16 = 0 (x − 16)(x − 1) = 0 x_1 = 16; x_2 = 1 Second eq. (√x) − 2 = 4(√x) − x − 2 x = 3(√x) x_3 = 9 x ∈ {1, 9} ⇒ sum = 10

$$\:\mid\sqrt{{x}}\:−\:\mathrm{2}\mid\:=\:\mathrm{4}\sqrt{{x}}\:−\:{x}\:−\:\mathrm{2} \\ $$ $$\:\:\sqrt{{x}}\:−\:\mathrm{2}\:=\:\pm\:\left(\mathrm{4}\sqrt{{x}}\:−\:{x}\:−\:\mathrm{2}\right) \\ $$ $$\:{First}\:{eq}. \\ $$ $$\:\:\sqrt{{x}}−\:\mathrm{2}\:=\:−\left(\mathrm{4}\sqrt{{x}}\:−\:{x}\:−\:\mathrm{2}\right) \\ $$ $$\:\sqrt{{x}}\:−\:\mathrm{2}\:=\:{x}\:+\:\mathrm{2}\:−\mathrm{4}\sqrt{{x}} \\ $$ $$\:\mathrm{5}\sqrt{{x}}\:=\:{x}\:+\:\mathrm{4} \\ $$ $$\:\mathrm{25}{x}\:=\:{x}^{\mathrm{2}} \:+\:\mathrm{16}\:+\:\mathrm{8}{x} \\ $$ $$\:{x}^{\mathrm{2}} \:−\:\mathrm{17}{x}\:+\:\mathrm{16}\:=\:\mathrm{0} \\ $$ $$\:\left({x}\:−\:\mathrm{16}\right)\left({x}\:−\:\mathrm{1}\right)\:=\:\mathrm{0} \\ $$ $$\:{x}_{\mathrm{1}} \:=\:\mathrm{16};\:{x}_{\mathrm{2}} \:=\:\mathrm{1} \\ $$ $$\:{Second}\:{eq}. \\ $$ $$\:\sqrt{{x}}\:−\:\mathrm{2}\:=\:\mathrm{4}\sqrt{{x}}\:−\:{x}\:−\:\mathrm{2} \\ $$ $$\:{x}\:=\:\mathrm{3}\sqrt{{x}} \\ $$ $$\:{x}_{\mathrm{3}} \:=\:\mathrm{9}\: \\ $$ $$\:{x}\:\in\:\left\{\mathrm{1},\:\mathrm{9}\right\}\:\:\Rightarrow\:{sum}\:=\:\mathrm{10} \\ $$

Commented bymanxsol last updated on 04/Dec/22

analisis inicial (√x)−4⟨0⇒0⟨(√x)⟨4⇒0⟨x⟨16 two parts 0⟨(√x)⟨2 2≪(√x)⟨4 −2⟨(√x)−2⟨0 0≪(√x)−2⟨4

$${analisis}\:{inicial} \\ $$ $$\sqrt{{x}}−\mathrm{4}\langle\mathrm{0}\Rightarrow\mathrm{0}\langle\sqrt{{x}}\langle\mathrm{4}\Rightarrow\mathrm{0}\langle{x}\langle\mathrm{16} \\ $$ $${two}\:{parts} \\ $$ $$\mathrm{0}\langle\sqrt{{x}}\langle\mathrm{2}\:\:\:\:\:\mathrm{2}\ll\sqrt{{x}}\langle\mathrm{4} \\ $$ $$−\mathrm{2}\langle\sqrt{{x}}−\mathrm{2}\langle\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{0}\ll\sqrt{{x}}−\mathrm{2}\langle\mathrm{4}\:\:\:\:\: \\ $$

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