Find all a,b,c ∈R that satisfy both equations: a+b+c=63 ab+bc+ac=2021

Question

Find all a,b,c ∈R that satisfy both equations:  a+b+c=63  ab+bc+ac=2021

Velsenw · Accepted Answer

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} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a

Rewrite the formula as follows.

{(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)

{(a + b + c)}^{2} - 2 (a b + b c + c a) = a^{2} + b^{2} + c^{2}

a^{2} + b^{2} + c^{2} = {(a + b + c)}^{2} - 2 (a b + b c + c a)

Substitute a+b+c=63 and ab+bc+ac=2021 in the above formula to find the unknowns if exists.

a^{2} + b^{2} + c^{2} = {(63)}^{2} - 2 (2021)

=3969-4042

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,

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.

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

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