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

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