Vieta's formulas

Vieta's formulas relate the roots of a quadratic equation $ax^2+bx+c=0$ to its coefficients — without solving the equation:

$x_1+x_2=-\frac{b}{a}\qquad x_1\cdot x_2=\frac{c}{a}$

They hold even when $\Delta<0$ (giving the sum and product of the complex roots). They are handy for quickly checking solutions, finding roots "by inspection", and problems where you need not compute the discriminant.

Conversely — given the roots, you can rebuild the equation (for $a=1$): $x^2-(x_1+x_2)x+x_1 x_2=0$. The calculator works both ways.