Development of the Quadratic Formula, with Explanatory Notes

By Mike Turnbull

Table of Contents

1. Introduction
2. What is a Mathematical Formula
3. Derivation of The Quadratic Formula

Introduction

A quadratic equation (often referred to suscinctly as a quadratic) is any equation that can be rearranged into the following form:

where $x$ represents an unknown value, and $a$ , $b$ , and $c$ each represent coefficients of known values, where $a\ne 0$ .

If $a=0$ and $b\ne 0$ then the equation is linear, not quadratic.

If $a=0$ and $b=0$ and $c\ne 0$ then the equation is a constant, not linear, nor quadratic.

The coefficients $a$ , $b$ , and $c$ may be referred to respectively as, the quadratic coefficient, the linear coefficient and the constant coefficient.

The name for a quadratic expression comes from quadratum, a Latin word which can have the English meanings: square, quadrature, quadrangle, or four-sided. These meanings, which all relate to objects having four sides, or four distinguishing features, somewhat obscure the use of the word quadratic as it is used when referring to single variable polynomials of order two. However, since the area of a quadralateral is obtained from the product of the length of its two sides - which, for a square is simply the product of the same length, or the square of the side length - the product of the same value by itself has traditionally been called squaring the number. Since the degree of a quadratic is 2, meaning that the highest order variable is squared, any single variable polynomial of order two is referred to as a quadratic polynomial; or, more suscinctly, as a quadratic.

For instance, the following equation is a quadratic.

The above quadratic, when graphed over the range of values from $x=-2$ to $x=\mathrm{+4}$ , has the shape shown in Figure 1.

This characteristic curved shape is known as a parabola.

Quadratic equations are found in mathematics concerning many areas of physics and engineering. For instance, one important use of quadratic equations is to model balistic projectile motion. This is the trajectory that a bullet or artillary shell will take after it leaves the rifle or gun barrel (if we ignore wind resistance). This really is rocket science!

Of particular interest are the values of $x$ for which the equation evaluates to zero. Inspection of Figure 1 indicates that, for that particular equation (and this is also true in general, but may not be so in all cases) there are two values of $x$ which produce a $y$ value of zero.

In Figure 1 this situation occurs when $x\approx -1.33$ and $x\approx \mathrm{+3.0}$ . It's difficult to say that these values are accurate by just viewing the graph; however, in practical applications it is highly desirable to determine these values as accurately and precisely as possible. Consequently just judging the values from a graph is not good enough. What is needed is an algebraic formula for determining the values precisely - and that is where the quadratic formula is used.

What is a Mathematical Formula

Many of us may have read a mystery novel, or seen a movie, about the search for a secret chemical formula that is going to save the World (or blow it up). This chemical formula is a recipe that explaines how to mix various chemicals, in precise proportions, in order to achieve the end result - hopefully it also containes detailed notes on how to NOT blow ourselves up in the mixing process.

Many of us have also been deluged with the Internet buzzword of the decade, the all-powerful algorithm; especially as it is misused to explain how the Google search engine works by using its secret search algorithm, when, in fact, it doesn't use an algorithm at all - it uses a formula.

So - what is the difference between a formula and an algorithm?

Well, it is true that an algoritm is a type of formula, but not the other way around; a formula is not necessarily an algorithm.

The term algorithm was originally used to refer to an ordered list of algebraic manipulations that need to be carried out in order to perform a particular mathematical calculation. You should note that the words algebra and algorithm stem from the same etymological root (in this case an Arabic root). This is no coincidence.

Over the relatively recent years the word algorithm has been usurped by people who know no different to refer to any method of achieving a particular goal. This started in the computer programming industry where patterns of computer application code started to be referred to as algorithms. At the beginning this referred to computer code that performed mathematical calculations, but quickly morphed onto code that really did not perform mathematicam calculations at all. The first of such computer algorithms was code that performed the sorting of lists of numerical or textual items into numerical or alphabetical order.

A good example of what a mathematical algorithm is, is the steps used to add two numbers in order to calculate their sum. Similarly, the steps used to subtract, multiply, or divide two numbers are true algorithms; or to square a number, or find its square root.

Examples of what are NOT algorithms are the sorting of the words in a Shakespearean play, or asking Google who is the current richest person on Earth.

A mathematical formula is a mathematical (usually an algebraic) equation that provides a recipe for carrying out a desired mathematical operation that will result in a solution for a mathematical question.

For instance, we may ask, "What values for the variable $x$ will satisfy the following algebraic equation?"

$$a{x}^2+bx+c=0$$

The answer to this question is a mathematical formula that will describe how to calculate the desired values for $x$ .

This formula will be written as another algebraic equation that encapsulates the steps required to be performed in order to solve the problem posed in the question; that is, in order to solve the problem; or, to say it another way, to find a solution to the problem. Some of those steps may indeed require well known algorithms to be followed in performing calculations.

SPOILER ALERT! The answer to the above question is:

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

This general solution shown as Equation 2, for the question posed in Equation 1, is known as The Quadratic Formula. It covers all cases, including all real and complex solutions, and was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing the quadratic formula in the formulation shown above.

In what follows we are going to go step-by-step through the derivation of The Quadratic Formula; However, if you haven't studied the wab page titled Completing the Square yet, you should do so now.

Derivation of The Quadratic Formula

We start by repeating Equation 1 as Equation 3, with the added necessary condition that the value represented by the quadratic coefficient $a$ cannot be equal to zero.

$$a{x}^2+bx+c=0\text{where}a\ne 0$$

It is noted here that the three coefficients are real variables of indeterminate sign and absolute numerical values; and, therefore, the rules pertaining to The Square Root of the Square of Real Variables of Unknown Polarity and Absolute Numerical Value will apply in the following algebraic development.

However, since $a\ne 0$ We can divide both sides by $a$ to give the following.

Now subtract $\frac{c}{a}$ from both sides, to give:

The left hand side can now have the technique of Completing the Square applied to it. This is done by adding the square of half the coefficient of $x$ to both sides, as follows.

It is noted here that the term being added to both sides contains the factor $2$ in its denominator, NOT $-2$ , and that, therefore, the rules pertaining to The Square Root of the Square of Real Values of Known Polarity will apply in the following algebraic development.

It is also noted that the variables $b$ in the numerator, and the $a$ in the denominator, as previously noted, are real variables of indeterminate sign and absolute numerical values; and, therefore, the rules pertaining to The Square Root of the Square of Real Variables of Unknown Polarity and Absolute Numerical Value will apply in the following algebraic development.

We now do some tidying up to ensure that convenient coefficients are produced to assist further development, as follows.

Which leads to:

We now take the square root of both sides, keeping in mind that the square root of the numerator on the right hand side could be positive or negative.

We now subtract $\frac{b}{2a}$ from both sides to give the final solution as follows.

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