# Deriving the Quadratic Equation From the Roots Up

Solving Quadratics using gradients at the roots

An important characteristic of the Quadratic Polynomial ** y=ax²+bx+c **is that it retains its parabolic

**shape and size**

*ax²***wherever it is located in the x-y grid. The other terms,**

*are just the,*

**bx +c***“Sit where we say”*, ushers!

This architectural symmetry and stability means that the gradients at the roots are of equal magnitude and opposite sign and can be used to calculate the roots.

For example, we will show how a quadratic such as ** y=2x²+x-3 **has root gradients:

giving** dy/dx=+-5**, and since

*we can transpose and find*

**dy/dx=4x+1**,**and**

*x=+1***at the roots!**

*-1.5*Accordingly we can proceed to construct the standard Quadratic Equation from the roots up!

This post assumes year 10 or 11 level high school math with the following refresher.

## T**he Standard Quadratic Equation**

To begin, let’s remind ourselves of a Quadratic Polynomial, which typically takes the form:

In this, all three terms, ** ax²**,

**and**

*bx***,**

*c***combine to construct each pixel of the parabolic component**

*, in the*

**ax²***x-y*grid of its graph.

For a detailed review refer to *Maths is Graphs — A Visual Perspective*.

The roots can be found by use of the standard Quadratic Equation shown below.

## C**onstruction From the Roots Up**

Let ** y=ax²+bx+c **hence:

** dy/dx=2ax+b**,

**transposing;**

**.**

*x=(dy/dx-b)/2a*Substituting for ** x **in the function

*y=ax²+bx+c:**y=(dy/dx-b)²/4a+b(dy/dx-b)/2a+c=(dy/dx)²-b²+4ac*

Hence, when ** y=0** at the roots:

as illustrated in Graph 1:

Since * dy/dx=2ax+b,* we can substitute for

**and write:**

*dy/dx*Giving the standard Quadratic Equation:

## E**xample: Finding Roots From Gradients**

Consider the following Polynomial,** y=x²-x-5, **shown in Graph 2 below with derivative

**shown at the roots.**

*dy/dx=2x-1*Example: *y=x²-x-5*

Giving: ** Root 2=2x-1=+4.58; x=5.58/2=2.79 **and

*Root 1=2x-1=-4.58=-1.79*

This post has presented an alternative method for developing the standard Quadratic Equation using the gradients at the roots. It can be intuitively useful in cases where the gradients are an algorithmic design objective and also simply recognising they are the very familiar discriminant.

I hope this post has helped you understand the math behind the formulas and graphs and the Quadratic Equation in a new way. Once again, I have tried to adopt a ‘visual and intuitive’ approach, making math work for you, not you for it!