Derivation of the quadratic formula

A computer scientist’s approach

Contents

1. Introduction
2. Completing the square
3. The derivation
4. Discussion
5. Endnotes
6. Works cited

Introduction

This article will just be a quick proof of the quadratic formula,¹ the formula that is used to work out the solutions to the general quadratic equation.² I will not be covering the geometric intuition of the derivation and the reader need only be familiar with methods in elementary algebra.

Completing the square

The method of completing the square is central to deriving the quadratic formula taking the form of the trinomial below (Eq. 1):

The following³ is an informal description of the procedure of completing the square:

1. Divide all coefficients in the trinomial by the a coefficient.
2. Transpose the c coefficient to the other side of the equation.
3. Multiply the b coefficient by ½ and store the value in a new variable called ε.
4. Factor the trinomial expression — note that this should always result in the form: (x + ε)²