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Question Number 120229 by Algoritm last updated on 30/Oct/20

Answered by benjo_mathlover last updated on 30/Oct/20

(√(36(x^2 +1)−x)) +(√(x^2 +36(x+1))) = x^2 (√(36([x+1]^2 −2x)−x)) +(√(x^2 +36(x+1))) = x^2 (√(36(x+1)^2 −73x)) +(√(x^2 +36(x+1))) = x^2 set (x+1)=u ⇒x=u−1 (√(36u^2 −73(u−1))) +(√((u−1)^2 +36u)) = (u−1)^2

$$\sqrt{\mathrm{36}\left({x}^{\mathrm{2}} +\mathrm{1}\right)−{x}}\:+\sqrt{{x}^{\mathrm{2}} +\mathrm{36}\left({x}+\mathrm{1}\right)}\:=\:{x}^{\mathrm{2}} \\ $$$$\sqrt{\mathrm{36}\left(\left[{x}+\mathrm{1}\right]^{\mathrm{2}} −\mathrm{2}{x}\right)−{x}}\:+\sqrt{{x}^{\mathrm{2}} +\mathrm{36}\left({x}+\mathrm{1}\right)}\:=\:{x}^{\mathrm{2}} \\ $$$$\sqrt{\mathrm{36}\left({x}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{73}{x}}\:+\sqrt{{x}^{\mathrm{2}} +\mathrm{36}\left({x}+\mathrm{1}\right)}\:=\:{x}^{\mathrm{2}} \\ $$$${set}\:\left({x}+\mathrm{1}\right)={u}\:\Rightarrow{x}={u}−\mathrm{1} \\ $$$$\sqrt{\mathrm{36}{u}^{\mathrm{2}} −\mathrm{73}\left({u}−\mathrm{1}\right)}\:+\sqrt{\left({u}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{36}{u}}\:=\:\left({u}−\mathrm{1}\right)^{\mathrm{2}} \\ $$

Answered by MJS_new last updated on 30/Oct/20

x≤−6(3+2(√2))∨x≥6(−3+2(√2)) ⇔ x≤≈−34.97∨x≥≈−1.029 both sides ≥0 ⇒ squaring and trsnsforming 2(√(36x^2 −x+36))(√(x^2 +36x+36))=x^4 −37x^2 −35x−72 squaring and transforming again (⇒ false solutions!) x^2 (x^6 −74x^4 −70x^3 +1081x^2 −2590x+1369)=0 x_(1, 2) =0 ⇒ false x_3 ≈.7351 false x_4 ≈8.43508 the 4 complex solutions are false ⇒ x≈8.43508

$${x}\leqslant−\mathrm{6}\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)\vee{x}\geqslant\mathrm{6}\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right) \\ $$$$\Leftrightarrow\:{x}\leqslant\approx−\mathrm{34}.\mathrm{97}\vee{x}\geqslant\approx−\mathrm{1}.\mathrm{029} \\ $$$$\mathrm{both}\:\mathrm{sides}\:\geqslant\mathrm{0}\:\Rightarrow\:\mathrm{squaring}\:\mathrm{and}\:\mathrm{trsnsforming} \\ $$$$\mathrm{2}\sqrt{\mathrm{36}{x}^{\mathrm{2}} −{x}+\mathrm{36}}\sqrt{{x}^{\mathrm{2}} +\mathrm{36}{x}+\mathrm{36}}={x}^{\mathrm{4}} −\mathrm{37}{x}^{\mathrm{2}} −\mathrm{35}{x}−\mathrm{72} \\ $$$$\mathrm{squaring}\:\mathrm{and}\:\mathrm{transforming}\:\mathrm{again} \\ $$$$\left(\Rightarrow\:\mathrm{false}\:\mathrm{solutions}!\right) \\ $$$${x}^{\mathrm{2}} \left({x}^{\mathrm{6}} −\mathrm{74}{x}^{\mathrm{4}} −\mathrm{70}{x}^{\mathrm{3}} +\mathrm{1081}{x}^{\mathrm{2}} −\mathrm{2590}{x}+\mathrm{1369}\right)=\mathrm{0} \\ $$$${x}_{\mathrm{1},\:\mathrm{2}} =\mathrm{0}\:\Rightarrow\:\mathrm{false} \\ $$$${x}_{\mathrm{3}} \approx.\mathrm{7351}\:\mathrm{false} \\ $$$${x}_{\mathrm{4}} \approx\mathrm{8}.\mathrm{43508} \\ $$$$\mathrm{the}\:\mathrm{4}\:\mathrm{complex}\:\mathrm{solutions}\:\mathrm{are}\:\mathrm{false} \\ $$$$\Rightarrow \\ $$$${x}\approx\mathrm{8}.\mathrm{43508} \\ $$

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