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Question Number 202250 by MATHEMATICSAM last updated on 23/Dec/23

(a^2  − bc)x^2  + 2(b^2  − ca)x + (c^2  − ab) = 0  has two equal roots. Show that either   b = 0 or (a^2 /(bc)) + (b^2 /(ca)) + (c^2 /(ab)) = 3.

(a2bc)x2+2(b2ca)x+(c2ab)=0hastwoequalroots.Showthateitherb=0ora2bc+b2ca+c2ab=3.

Answered by esmaeil last updated on 23/Dec/23

r_1 −r_2 =^(r_1 =r_2 ) ((√δ^( ′) )/(∣a∣))=0→  ((√((b^2 −ca)^2 −(c^2 −ab)(a^2 −bc)))/(a^2 −bc))=0→  b^4 +c^2 a^2 −2b^2 ac−c^2 a^2 +bc^3 +ba^3 −ab^2 c  =0  →b^4 −3b^2 ac+bc^3 +ba^3 ==→  b(b^3 −3bac+c^3 +a^3 )=0→  b=0 or (b^3 /(abc))−((3abc)/(abc))+(c^3 /(abc))+((ba^3 )/(abc))=0→  (b^2 /(ac))+(c^2 /(ab))+(a^2 /(bc))=3

r1r2=r1=r2δa=0(b2ca)2(c2ab)(a2bc)a2bc=0b4+c2a22b2acc2a2+bc3+ba3ab2c=0b43b2ac+bc3+ba3==→b(b33bac+c3+a3)=0b=0orb3abc3abcabc+c3abc+ba3abc=0b2ac+c2ab+a2bc=3

Answered by Rasheed.Sindhi last updated on 23/Dec/23

α & α are the roots  α+α=2α=−((2(b^2  − ca))/((a^2  − bc)))⇒α=((b^2  − ca)/(a^2  − bc))  α^2 =(((b^2  − ca)/(a^2  − bc)))^2 =((c^2  − ab)/(a^2  − bc))        (((b^2  − ca)^2 )/(a^2  − bc))=c^2  − ab       (b^2  − ca)^2 =(a^2  − bc)(c^2  − ab)   b^4 −2ab^2 c+c^2 a^2 =c^2 a^2 −a^3 b−bc^3 +ab^2 c    b^4 −3ab^2 c=−a^3 b−bc^3     b^4 −3ab^2 c+a^3 b+bc^3 =0  b(b^3 −3abc+a^3 +c^3 )=0  b=0✓ ∣ a^3 +b^3 +c^3 =3abc                  ((a^3 +b^3 +c^3 )/(abc))=3               (a^2 /(bc)) + (b^2 /(ca)) + (c^2 /(ab)) = 3 ✓

α&αaretherootsα+α=2α=2(b2ca)(a2bc)α=b2caa2bcα2=(b2caa2bc)2=c2aba2bc(b2ca)2a2bc=c2ab(b2ca)2=(a2bc)(c2ab)b42ab2c+c2a2=c2a2a3bbc3+ab2cb43ab2c=a3bbc3b43ab2c+a3b+bc3=0b(b33abc+a3+c3)=0b=0a3+b3+c3=3abca3+b3+c3abc=3a2bc+b2ca+c2ab=3

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