Find all a,b,c $\in \mathbb{R}$ that satisfy both equations:

a+b+c=63

ab+bc+ac=2021

a+b+c=63

ab+bc+ac=2021

York
2021-05-10
Answered

Find all a,b,c $\in \mathbb{R}$ that satisfy both equations:

a+b+c=63

ab+bc+ac=2021

a+b+c=63

ab+bc+ac=2021

You can still ask an expert for help

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.

${(a+b+c)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2ab+2bc+2ca$

Rewrite the formula as follows.

${(a+b+c)}^{2}={a}^{2}+{b}^{2}+{c}^{2}+2(ab+bc+ca)$

$(a+b+c)}^{2}-2(ab+bc+ca)={a}^{2}+{b}^{2}+{c}^{2$

${a}^{2}+{b}^{2}+{c}^{2}={(a+b+c)}^{2}-2(ab+bc+ca)$

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

${a}^{2}+{b}^{2}+{c}^{2}={\left(63\right)}^{2}-2\left(2021\right)$

=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.

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.

Rewrite the formula as follows.

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

=3969-4042

It is known that, the square of any real number value is positive and obviously the sum of its squares also positive.

Note that,

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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