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Question Number 202250 by MATHEMATICSAM last updated on 23/Dec/23
(a2−bc)x2+2(b2−ca)x+(c2−ab)=0hastwoequalroots.Showthateitherb=0ora2bc+b2ca+c2ab=3.
Answered by esmaeil last updated on 23/Dec/23
r1−r2=r1=r2δ′∣a∣=0→(b2−ca)2−(c2−ab)(a2−bc)a2−bc=0→b4+c2a2−2b2ac−c2a2+bc3+ba3−ab2c=0→b4−3b2ac+bc3+ba3==→b(b3−3bac+c3+a3)=0→b=0orb3abc−3abcabc+c3abc+ba3abc=0→b2ac+c2ab+a2bc=3
Answered by Rasheed.Sindhi last updated on 23/Dec/23
α&αaretherootsα+α=2α=−2(b2−ca)(a2−bc)⇒α=b2−caa2−bcα2=(b2−caa2−bc)2=c2−aba2−bc(b2−ca)2a2−bc=c2−ab(b2−ca)2=(a2−bc)(c2−ab)b4−2ab2c+c2a2=c2a2−a3b−bc3+ab2cb4−3ab2c=−a3b−bc3b4−3ab2c+a3b+bc3=0b(b3−3abc+a3+c3)=0b=0✓∣a3+b3+c3=3abca3+b3+c3abc=3a2bc+b2ca+c2ab=3✓
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