# Is the quadratic formula related to the process of completing the square?

jeew-m | Certified Educator

yes it is related to completing the square. In the process of deriving quadratic formulae we come through completing the square. In the following steps it will be shown.

As you know if you have a quadratic equation `ax^2+bx+c = 0` then by quadratic formulae you say that the roots of the equation will be;

`x = (-b+-sqrt(b^2-4xxaxxc))/(2xxa)`

Now let us start `ax^2+bx+c = 0` from completing the square.

`ax^2+bx+c = 0`

`x^2+(b/a)x+c/a = 0`

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

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

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

up to this point it is completing the square. From here onwards we will derive the quadratic equation.

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

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

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

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

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

This is the roots from quadratic equation that derived through completing square.