Quadratic equation/R/Solution formula/Fact

Solution formula for quadratic equations

Let

${}aX^{2}+bX+c=0\,$

denote a real quadratic equation.

Then the following holds.

1. For

   ${}b^{2}-4ac<0\,,$

   there is no real solution.

2. For

   ${}b^{2}-4ac=0\,,$

   the only solution is

   ${}x={\frac {-b}{2a}}\,$

3. For

   ${}b^{2}-4ac>0\,,$

   there exist the two solutions

   ${}x_{1,2}={\frac {\pm {\sqrt {b^{2}-4ac}}-b}{2a}}\,.$

Write a proof