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.

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.