Quadratic equation

The quadratic equation is a formula for finding the roots of a second-degree polynomial. Any second-degree polynomial in the variable $x$ will be of the form

$ax^{2}+bx+c$

where $a$, $b$, and $c$ are real constants and $a$ is not zero (if it was, the polynomial would only be first-degree). The roots of the polynomial are the particular values of $x$ for which the polynomial equals zero. The Fundamental Theorem of Algebra tells us that we should expect there to be two roots for a second-degree polynomial, although they might be equal in some cases. If we call the roots $x_{+}$ and $x_{-}$ then what we are saying is that

$ax_{\pm }^{2}+bx_{\pm }+c=0\ .$

This is where the quadratic equation comes in. It tells us that the solutions $x_{+}$ and $x_{-}$ can always be found from the equation

$x_{\pm }={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\ .$

Proof

The simplest way to show that the values $x_{\pm }$ are in fact roots to the polynomial above is to substitute them into the equation

${\begin{aligned}ax_{\pm }^{2}+bx_{\pm }+c&=a\left({\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\right)^{2}+b{\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}+c\\&={\frac {1}{4a}}\left(b^{2}\pm 2b{\sqrt {b^{2}-4ac}}+b^{2}-4ac\right)-{\frac {b^{2}\pm b{\sqrt {b^{2}-4ac}}}{2a}}+c\\&={\frac {b^{2}\pm b{\sqrt {b^{2}-4ac}}}{2a}}-c-{\frac {b^{2}\pm b{\sqrt {b^{2}-4ac}}}{2a}}+c\\&=0\ ,\end{aligned}}$

as desired.