Find all a,b,c \in \mathbb{R} that satisfy both equations: a+b+c=63 ab+bc+ac=2021

York 2021-05-10 Answered
Find all a,b,c R that satisfy both equations:
a+b+c=63
ab+bc+ac=2021
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Expert Answer

Velsenw
Answered 2021-05-12 Author has 91 answers
Step 1
The given equations are a+b+c=61 and ab+bc+ac=2021.
The objective is to find the real values of a, b, c that satisfy the above equations.
Step 2
The Square of a Trinomial formula stated as follows.
(a+b+c)2=a2+b2+c2+2ab+2bc+2ca
Rewrite the formula as follows.
(a+b+c)2=a2+b2+c2+2(ab+bc+ca)
(a+b+c)22(ab+bc+ca)=a2+b2+c2
a2+b2+c2=(a+b+c)22(ab+bc+ca)
Substitute a+b+c=63 and ab+bc+ac=2021 in the above formula to find the unknowns if exists.
a2+b2+c2=(63)22(2021)
=3969-4042
a2+b2+c2=73
It is known that, the square of any real number value is positive and obviously the sum of its squares also positive.
Note that, a2+b2+c2=73. That is, the sum of squares of the required numbers are negative.
Therefore, there is no such real number exist that satisfy the equations a+b+c=63 and ab+bc+ac=2021.
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