$0\=$

$a x^2\+b x\+c$

$\=$

$a\cdot \left(x^2\+\frac{b}{a}x\right)\+c$

$\=$

$a\cdot \left(x^2\+\frac{b}{a}x\+{\left(\frac{b}{2 a}\right)}^2\right)\+c-a\cdot {\left(\frac{b}{2 a}\right)}^2$

$\=$

$a\cdot {\left(x\+\frac{b}{2 a}\right)}^2\+c-\frac{b^2}{4 a}$

so

$a\cdot {\left(x\+\frac{b}{2 a}\right)}^2\=$

$\frac{b^2}{4 a}-c$

${\left(x\+\frac{b}{2 a}\right)}^2\=$

$\frac{b^2}{4 a^2}-\frac{c}{a}$

and finally

$x\=$

$-\frac{b}{2 a}\pm \sqrt{\frac{b^2}{4 a^2}-\frac{c}{a}}$

$\=$

$\frac{-b\pm \sqrt{b^2-4 a c}}{2 a}$.