Deriving the Quadratic Formula (the easy way)
May 17, 2011 6 Comments
In Bowen’s post from April 25, he showed a great method for factoring non-monic quadratics. Using that same method, you can derive the quadratic formula pretty cleanly, without lots of fractions, rationalizations, and the like.
The goal is to find a solution to the generic quadratic equation:
We can’t factor using sums and products, so we resort to completing the square. The equation would be much easier to work with if it were monic. We could divide through by , but then there’s lots of fractions to keep track of. Instead, let’s multiply both sides by and make it a quadratic in :
Hmmm. Again, things would be so much easier to complete the square if that middle term were even. We can multiply both sides by 2 to do that, but then the first term isn’t a perfect square. Fine, let’s multiply both sides by 4 then:
Now, just rewrite that a little bit:
and now we can put our fingers over and see this as a simpler monic:
And now, complete the square. First, let’s put that constant on the other side:
To get a complete square on the left side, we need a , so add it to each side:
The left side is now a perfect square… and the right side looks familiar… Let’s factor the left side:
To solve for , take the square root of each side:
and subtract from each side:
What was again? Lift that finger…oh, yeah, :
So finish this off by dividing each side by :
And there you have it… the quadratic formula, with no fractions until the final step. Enjoy!