Many remember the quadratic formula:

Few, however, remember how it was derived. How do you get from a general quadratic equation, to the general solution shown above?

First,
let's recall that the square of a sum can be expressed as the square
of the first plus twice the product plus the square of the second,
.
We want to manipulate the quadratic equation to get a similar
expression where *a* =
*x*. Let's
start by getting rid of the coefficient of *x*^{2}.

Now we move the last term to the other side of the equal sign in preparation for adding another term that will make the expression a perfect square.

But
what term can we add to make the left expression a perfect square?
Our second term needs to be twice the product of something and *x*.

It should now be apparent that we have to add the square of to make the left expression a perfect square.

Expressing the left as a square and combining the fractions on the right gives:

Now we can take the square root of both sides to get:

which simplifies to

Now
we're nearly done. Solving for *x*
gives: