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If-a-2-bc-b-2-ac-c-2-ab-is-AP-where-a-c-12-find-the-value-of-a-b-c-

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Question Number 103620 by bobhans last updated on 16/Jul/20
If a^2 −bc, b^2 −ac, c^2 −ab is AP where a+c = 12, find the value of a+b+c
Ifa2−bc,b2−ac,c2−abisAPwherea+c=12,findthevalueofa+b+c
Answered by bemath last updated on 16/Jul/20
⇒AP : 2(b^2 −ac)=a^2 −bc+c^2 −ab 2b^2 −2ac = a^2 +c^2 −ab−bc 2b^2 −2ac = (a+c)^2 −2ac−ab−bc 2b^2 = 144−ab−b(12−a) 2b^2 = 144−12b b^2 +6b−72 = 0 ⇒(b+12)(b−6)=0→ { ((b=6)),((b=−12)) :} ∴ a+b+c = 18 or zero
⇒AP:2(b2−ac)=a2−bc+c2−ab2b2−2ac=a2+c2−ab−bc2b2−2ac=(a+c)2−2ac−ab−bc2b2=144−ab−b(12−a)2b2=144−12bb2+6b−72=0⇒(b+12)(b−6)=0→{b=6b=−12∴a+b+c=18orzero

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