`ax^2+bx+c=0`

This is a general quadratic equation.

Rearrange:

`ax^2+bx=-c`

Divide throughout by `a`:

`x^2+b/a x =-c/a`

Write as a perfect square:

`x^2 + b/a x +(b/(2a))^2=-c/a+(b/(2a))^2`

`(x+b/(2a))^2=(-4ac+b^2)/(4a^2)`

Solve:

`x+b/(2a)= +-sqrt(-4ac+b^2)/(2a)`

`x=-b/(2a)+-sqrt(b^2-4ac)/(2a)`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

We'll use this result a great deal throughout the rest of the math we study.