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if-a-b-c-3-and-a-3-b-3-c-3-27-find-a-2-b-2-c-2-

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Question Number 72027 by mathmax by abdo last updated on 23/Oct/19
if a+b+c =3 and a^3 +b^3 +c^3 =27 find a^2 +b^2 +c^2
ifa+b+c=3anda3+b3+c3=27finda2+b2+c2
Commented by Rasheed.Sindhi last updated on 26/Oct/19
Insufficient data for unique solution. There are three unknowns,so three equations are required for unique solution.
Insufficientdataforuniquesolution.Therearethreeunknowns,sothreeequationsarerequiredforuniquesolution.
Answered by Rasheed.Sindhi last updated on 26/Oct/19
For unique solution a third equation is necessary,because there are three unknowns. Since the third equation is not given we assume value of one unknown: Let c=1 a+b=2 , a^3 +b^3 =26 (a+b)^3 =8 a^3 +b^3 +3ab(a+b)=8 26+3ab(2)=8 ab=((−18)/6)=−3 a(2−a)=−3 a^2 −2a−3=0 (a−3)(a+1)=0 a=3 ∨ a=−1 b=2−3=−1 ∨ b=2+1=3 (a,b,c)=(3,−1,1) or (a,b,c)=(−1,3,1) If c=1,−1 or 3: a^2 +b^2 +c^2 =(−1)^2 +(3)^2 +(1)^2 =6
Foruniquesolutionathirdequationisnecessary,becausetherearethreeunknowns.Sincethethirdequationisnotgivenweassumevalueofoneunknown:Letc=1a+b=2,a3+b3=26(a+b)3=8a3+b3+3ab(a+b)=826+3ab(2)=8ab=−186=−3a(2−a)=−3a2−2a−3=0(a−3)(a+1)=0a=3∨a=−1b=2−3=−1∨b=2+1=3(a,b,c)=(3,−1,1)or(a,b,c)=(−1,3,1)Ifc=1,−1or3:a2+b2+c2=(−1)2+(3)2+(1)2=6

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