A quadratic expression is monic if the coefficient of its quadratic term is 1. Let f(x), g(x), and h(x) be monic quadratics such that the sum of the roots of f(x) is a, the sum of the root of g(x) is b, and the sum of the roots of h(x) is c. Find the sum of the solutions to f(x) + g(x) + h(x) = 0 in terms of a,b, and c.

Imnotamaster Jun 9, 2019

#1**+1 **

\(p(x) = x^2 + c_1 x + c_0 \Rightarrow r_1 + r_2 = -c_1\\ f(x) = x^2 - a x + f_0\\ g(x) = x^2 - b x+ g_0\\ h(x) = x^2 - c x + h_0\\ y(x) = f(x) + g(x) + h(x) = 3x^2 -(a+b+c)x + (f_0 +g_0+h_0)\)

\(y(x) = x^2 - \dfrac 1 3(a+b+c)x + \dfrac 1 3 (f_0 + g_0 + h_0)\\ \text{This is the same form as $p(x)$ listed above with $c_1=-\dfrac 1 3(a+b+c)\\$and thus the sum of the roots is $\dfrac 1 3 (a+b+c)$}\)

.Rom Jun 10, 2019