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b-c-x-2-c-a-x-a-b-0-if-the-eqation-roots-are-eqal-you-proved-that-2b-a-c-

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Question Number 26680 by Sr@2004 last updated on 28/Dec/17
(b−c)x^2 +(c−a)x+(a−b)=0 if the eqation roots are eqal you proved that 2b=a+c.
(b−c)x2+(c−a)x+(a−b)=0iftheeqationrootsareeqalyouprovedthat2b=a+c.
Answered by mrW1 last updated on 28/Dec/17
equal roots ⇒ Δ=0 Δ=(c−a)^2 −4(b−c)(a−b)=0 ⇒c^2 −2ac+a^2 −4(ab−ac−b^2 +bc)=0 ⇒c^2 +a^2 −4ab+2ac+4b^2 −4bc=0 ⇒(a+c)^2 −4(a+c)b+4b^2 =0 ⇒(a+c−2b)^2 =0 ⇒2b=a+c
equalroots⇒Δ=0Δ=(c−a)2−4(b−c)(a−b)=0⇒c2−2ac+a2−4(ab−ac−b2+bc)=0⇒c2+a2−4ab+2ac+4b2−4bc=0⇒(a+c)2−4(a+c)b+4b2=0⇒(a+c−2b)2=0⇒2b=a+c

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