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Tuesday, September 19, 2023

2023/034) Given a+b+c=3 and a^2+b^2+c^2 = 3 Solve for real a,b,c

We have 

a+b+c=3\cdots(1)

a^2+b^2+c^2=3\cdots(2)

Squaring (1) we get

a^2+b^2 + c^2 + 2ab + 2bc + 2ca = 9

Putting the value of a^2+b^2+c^2 from (2) we get

ab+bc+ca = 3

subracting above from (2) we get

a^2+b^2+c^2 - ab - bc - ca = 0

or 2a^2+2b^2+2c^2 - 2ab - 2bc - 2ca = 0

or (a^2-2ab + b^2) + (b^2-2bc+c^2) + (c^-2ca +a^2) = 0

or (a-b)^2 + (b-c)^2 + (c-a)^2= 0

above is possible only if a=b= c

putting in (1) we get a = b = c = 1

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