Answer to Question #265456 in Algebra for Pankaj

Question

If a, b and c are the roots of the equation x^3 -6x^2 +10x-6 =0 , find the values of a^2 +b^2 +c^2 and 1/a + 1/b + 1/c.

Answer 1

If $a, b$ and $c$ are the roots of the equation $x^3 -6x^2 +10x-6 =0 ,$ let us find the values of $a^2 +b^2 +c^2$ and $\frac{1}a + \frac{1}b + \frac{1}c.$

According to Vieta's formulas, $a+b+c=6,\ ab+ac+bc=10,\ abc=6.$

Then we get

$a^2 +b^2 +c^2=(a+b+c)^2-2(ab+bc+ac)=6^2-2\cdot 10=36-20=16$

and

$\frac{1}a + \frac{1}b + \frac{1}c=\frac{bc+ac+ab}{abc}=\frac{10}{6}=\frac{5}3.$