A Simple Derivation of the Quadratic Formula

In this blog post, I will talk about the famous quadratic formula - a formula for finding the zero(s) of a polynomial equation of degree 2. I first learned this equation in Algebra, but I had always thought of it as a mathematical truth, without actually knowing how it came to be or why it's justified. Now, I have the tools to show that it's correct and to actually derive it using simple algebra.


The quadratic formula is a formula used to solve for the zero(s) an arbitrary polynomial in the form $y = ax^2+bx+c$.  Here it is in it's standard form:

$$ x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} $$

My proof of this formula relies on an algebraic trick known as completing the square, which is another useful technique to solving equations of this form. While I personally prefer the completing the square method because it's faster, many people prefer using the quadratic formula because it doesn't require much thought or intuition; it's straight plug-and-chug. Here is my proof:

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