why do we have to complete the quare in order to get the quadratic formula

ok, i know how the formula is derived. but why did they have to complete the quare to get that equation? how fif they know that completing the square would give the right formula?

Because, in the standard quadratic equation - ax² + bx + c = 0 - there are two "x"s. To make x the subject of the equation, you have to make sure there is only one x written down. After you've done that, you just have to do a bit of rearanging to give the quadratic formula.
One way to get to a stage where you only have one "x" term is by completing the square.

With most other proofs, either a known method is used, or some genius insight has been had... or lots of trial and error! Note that while it may take thirty seconds to write down a proof of a result, somebody may well have taken thirty years to discover it!

yeah consider yourself lucky that a formula was found or else you will have to do it the long way..

They didn't know until it worked.

To add to what Dross said - in the general quadratic equation, you have an undefined variable x along with undefined parameters a, b and c. You can make these parameters whatever you want - they have no value at this stage! - so you might as well complete the square because it makes absolutely no difference to the values you later assign to the parameters.

x^2 - 2x -1 is always exactly the same as ((x-1)^2) -2. You've just changed the value of your constant by adding 2 and taking it away afterwards. It is a handy tip to remember, that you can always add anything you want to an equation as long as you take it away again at some point.

Integrating ln(x) runs along similar lines.


zaza