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 x2.
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: