# Product of Roots of Quadratic Equation

## Theorem

Let $P$ be the quadratic equation $a x^2 + b x + c = 0$.

Let $\alpha$ and $\beta$ be the roots of $P$.

Then:

$\alpha \beta = \dfrac c a$

## Proof

 $\ds \alpha$ $=$ $\ds \frac {-b + \sqrt {b^2 - 4 a c} } {2 a}$ Solution to Quadratic Equation $\ds \beta$ $=$ $\ds \frac {-b - \sqrt {b^2 - 4 a c} } {2 a}$ Without loss of generality, selecting $\alpha$ and $\beta$ as such $\ds \leadsto \ \$ $\ds \alpha \beta$ $=$ $\ds \frac {\paren {-b - \sqrt {b^2 - 4 a c} } \paren {- b + \sqrt {b^2 - 4 a c} } } {4 a^2}$ $\ds$ $=$ $\ds \frac {b^2 - \paren {b^2 - 4 a c} } {4 a^2}$ Difference of Two Squares $\ds$ $=$ $\ds \frac {4 a c} {4 a^2}$ $\ds$ $=$ $\ds \frac c a$

$\blacksquare$