How to simplify the following expression.
I would like to be able to simplify the following expression, if it is at all possible.
$$-\left(\frac{A}{3a} + \frac{B}{3b} + \frac{C}{3c}\right) \pm \frac{\sqrt{A^{2}(b^{2}c^{2}) + B^{2}(a^{2}c^{2}) + C^{2}(a^{2}b^{2}) - AB(abc^{2}) - AC(ab^{2}c) - BC(a^{2}bc)}}{3abc},$$
Thank you.
Edit
I believe that I can simplify it to the following, but I am not sure if this is the furthest that one can go to simplify the expression.
$$-\frac{1}{3}\left(\left(\frac{A}{a} + \frac{B}{b} + \frac{C}{c}\right) \mp \sqrt{\left(\frac{A^{2}}{a^{2}} + \frac{B^{2}}{b^{2}} + \frac{C^{2}}{c^{2}} - \frac{AB}{ab} - \frac{AC}{ac} - \frac{BC}{bc}\right)}\right),$$
$\endgroup$ 101 Answer
$\begingroup$HINT:You can go to the form $3X= -(\alpha + \beta+\gamma) \pm\sqrt{(\alpha + \beta + \gamma)^2 – (\alpha\beta + \alpha\gamma + \beta\gamma)}$ and this is actually six times the roots of
$X^2 +(\alpha + \beta+\gamma) X + \frac{(\alpha\beta + \alpha\gamma + \beta\gamma)}{4}=0$.
I fear you could not simplify without additional conditions on your letters $A,B,C,a,b,c$.
$\endgroup$ 2
