Question

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

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ANSWERED
asked 2021-05-10
Find all a,b,c \(\displaystyle\in{\mathbb{{{R}}}}\) that satisfy both equations:
a+b+c=63
ab+bc+ac=2021

Answers (1)

2021-05-12
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.
\(\displaystyle{\left({a}+{b}+{c}\right)}^{{{2}}}={a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}+{2}{a}{b}+{2}{b}{c}+{2}{c}{a}\)
Rewrite the formula as follows.
\(\displaystyle{\left({a}+{b}+{c}\right)}^{{{2}}}={a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}+{2}{\left({a}{b}+{b}{c}+{c}{a}\right)}\)
\(\displaystyle{\left({a}+{b}+{c}\right)}^{{{2}}}-{2}{\left({a}{b}+{b}{c}+{c}{a}\right)}={a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}\)
\(\displaystyle{a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}={\left({a}+{b}+{c}\right)}^{{{2}}}-{2}{\left({a}{b}+{b}{c}+{c}{a}\right)}\)
Substitute a+b+c=63 and ab+bc+ac=2021 in the above formula to find the unknowns if exists.
\(\displaystyle{a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}={\left({63}\right)}^{{{2}}}-{2}{\left({2021}\right)}\)
=3969-4042
\(\displaystyle{a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}=-{73}\)
It is known that, the square of any real number value is positive and obviously the sum of its squares also positive.
Note that, \(\displaystyle{a}^{{{2}}}+{b}^{{{2}}}+{c}^{{{2}}}=-{73}\). 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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