How do I prove that the sum of $x_1 + x_2$ and $x_1 \cdot x_2$ is rational?

Question

Given an equation $ax^2 + bx + c = 0$, with $a, b$ and $c$ being whole numbers, and this equation having two solutions. The solutions of such an equation are sometimes rational, but mostly not. How do you prove that the sum of the solutions $x_1 + x_2$ and the product $x_1 \cdot x_2$ are always rational?

Answer 1

From the comments, the idea here is to simply use the formula $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$$ from which it follows that $$x_1+x_2=-\frac ba,\quad x_1x_2=\frac ca.$$ These are called Vieta's formulas and can be generalised to polynomials of arbitrary degree.