Sum and Product of Roots

The quadratic formula
 

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that
 

where x₁ and x₂ are the roots of the quadratic equation ax² + bx + c = 0. The sum of roots x₁ + x₂ and the product of roots x₁·x₂ are common to problems involving quadratic equation.
 

Derivation of the Sum of Roots
$x_1 + x_2 = \dfrac{-b + \sqrt{b^2-4ac}}{2a} + \dfrac{-b - \sqrt{b^2-4ac}}{2a}$

$x_1 + x_2 = \dfrac{-b + \sqrt{b^2-4ac} - b - \sqrt{b^2-4ac} }{2a}$

$x_1 + x_2 = \dfrac{-2b}{2a}$
 

Derivation of the Product of Roots
$x_1 \, x_2 = \left( \dfrac{-b + \sqrt{b^2-4ac}}{2a} \right) \left( \dfrac{-b - \sqrt{b^2-4ac}}{2a} \right)$

By difference of two squares:
$x_1 \, x_2 = \dfrac{b^2 - (b^2-4ac)}{4a^2}$

$x_1 \, x_2 = \dfrac{4ac}{4a^2}$