Question
Prove that $\text{a}^2+\text{b}^2+\text{c}^2-\text{ab}-\text{bc}-\text{ca}$ is always non negetive for all values of $a, b$ and $c.$
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Answer
In the given problem, we have to prove $\text{a}^2+\text{b}^2+\text{c}^2-\text{ab}-\text{bc}-\text{ca}$ is always non negetive for all $a, b , c$ that is we have to prove that $\text{a}^2+\text{b}^2+\text{c}^2-\text{ab}-\text{bc}-\text{ca}\geq0$ Consider, $\text{a}^2+\text{b}^2+\text{c}^2-\text{ab}-\text{bc}-\text{ca}$
$\text{a}^2+\text{b}^2+\text{c}^2-\text{ab}-\text{bc}-\text{ca}$
$=\frac{1}{2}\big(2\text{a}^2+2\text{b}^2+2\text{c}^2-2\text{ab}-2\text{bc}-2\text{ca}\big)$
$=\frac{1}{2}[(\text{a}-\text{b}^2)+(\text{b}-\text{c}^2)+(\text{c}-\text{a}^2)]$
$\text{a}^2+\text{b}^2+\text{c}^2-\text{ab}-\text{bc}-\text{ca}$
$=\frac{1}{2}\big(2\text{a}^2+2\text{b}^2+2\text{c}^2-2\text{ab}-2\text{bc}-2\text{ca}\big)$
$=\frac{1}{2}[(\text{a}-\text{b}^2)+(\text{b}-\text{c}^2)+(\text{c}-\text{a}^2)]$
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