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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

36(x2+1)−x+x2+36(x+1)=x236([x+1]2−2x)−x+x2+36(x+1)=x236(x+1)2−73x+x2+36(x+1)=x2set(x+1)=u⇒x=u−136u2−73(u−1)+(u−1)2+36u=(u−1)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⩽−6(3+22)∨x⩾6(−3+22)⇔x⩽≈−34.97∨x⩾≈−1.029bothsides⩾0⇒squaringandtrsnsforming236x2−x+36x2+36x+36=x4−37x2−35x−72squaringandtransformingagain(⇒falsesolutions!)x2(x6−74x4−70x3+1081x2−2590x+1369)=0x1,2=0⇒falsex3≈.7351falsex4≈8.43508the4complexsolutionsarefalse⇒x≈8.43508

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