Completing the Square

Introduction

Proper understanding of what Algebra is, and how it works, relies on the student having a fundamental knowledge of the mathematical components that Algebra is comprised of. These Algebraic components are the basic tools that are used to explain and describe the functionality of those aspects of Algebra that they are especially developed to deal with.

The process of Competing the Square is one of those tools.

In Algebra there is no one unique way of expressing any particular mathematical statement - in fact, most Algebraic statements can be expressed in an infinity of ways. There are some Algebraic statements that can only be expressed in limited ways. A simple example of that is the addition of two variables, as follows.

x = a + b

An alternative way to express that same thing would be to reverse the order of the variables on the right hand side of the equation, as follows.

x = b + a

Another, rather trivial, alternate way of expressing the above equation would be to exchange the left and right hand side expressions, as follows.

b + a = x

or:

a + b = x

Each of the above variations are really expressing the same thing in slightly different trivial ways. Although some readers may prefer one variation over the others, none of the forms is incorrect. In fact, why we may want to use more than one expression in this instance is rather obscure. This is not to say that there may be good reasons for using the variations; but, when taken out of context like this, the exercise would seem to be just playing Algebraic games for fun. However, there are often very good resaons why we may want to transform an Algebraic expression from one form to another - principally:

To make the meaning of the expression easier to read and understand.
To elucidate the development of the expression.

For instance, which of the following equivalent Algebraic expressions do you think is easier to understand?

y = 6 x^{2} - 10 x - 24

or,

y = (3 x + 4) (2 x - 6)

In Algebra a monomial is an element consisting of a single numerical term. A polynomial is any Algebraic element that consists of more than one numerical term. A trinomial is a polynomial consisting of three terms.

The right hand side in the first of the above equations is expressed as a second order single variable trinomial or quadratic expression; whereas, that in the second equation is expressed as the product of two single order single variable binomials. These two expressions are equivalent, which can be demonstrated by performing the multiplication of the two single order binomials.

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.

Performing the multiplication of the two single order binomials is a relatively simple exercise; however, breaking the quadratic (the second order trinomial) down into its two single order binomial factors is not such a simple exercise. It could be done with a bit of trial and error, but that would only give you a result for that particular quadratic expression. If the constants, or the sign of the constants, were to be changed then we have a completely different quadratic expression - which would need a completely different process for factorisation. What is needed is a general formula for factorising any given quadratic expression.

That is where the technique of Completing the Square comes in handy.

In what follows the technique of Completing the Square will be explained. It is a technique that can later be used to develop a formula for factorising any quadratic expression or equation.

Completing the Square

Completing the Square is an algebraic technique for factoring quadratic expressions. It is often applied in any computation involving quadratic polynomials. Completing the square is accomplished by added to and subtracted from an expression the same value in order to rewrite it to incorporate the square of a binomial.

For example,

x^{2} + 2 x + 3

can be rewritten as

(x^{2} + 2 x + 1) + 3 - 1

which can then be rewritten as

{(x + 1)}^{2} + 2

The two expressions are totally equivalent, but the second one simplifies Algebraic manipulations in some situations.

Of course, the above example used a particular quadratic. It doesn't demonstrate how to proceede with a general quadratic, but it gives a hint to the direction needed to succeed.

We need to think a bit deeper about how to derive the value that is to be added and subtracted in order to form the perfect square of a binomial.

However, consider for a moment the following observation.

The number that we added and subtracted was half the value of the middle term of the original quadratic. Was this just fortuitous, or is there a pattern here?

Consider the following general quadratic expression.

a x^{2} + b x + c

This general form is different from our simple example (above) in a number of respects.

It contains the factor $a$ which multiplies the factor $x^{2}$ .

In our previous example this parameter was set to unity, thus making the transformation from $x^{2}$ to $x$ an uncomplicated matter of simply applying a square root. We need to get rid of the $a$ or at least shift it to somewhere else where it is more manageable.

This can be done as follows.

a (x^{2} + \frac{b}{a} x + \frac{c}{a})

This successfully isolates the quadatic with the desirable unity factor for the $x^{2}$ term, and reduces the complexity of further manipulation. We now only need to concentrate on the quadratic in parenthesis. We can pretty much ignore the $a$ factor momentarily, and just regard the $a b$ and $a c$ terms as new factors in the isolated quadratic.

Remember that we previously observed that the number we added and subtracted in order to form a perfect square was half the value of the middle term of the original quadratic, and we wondered whether this just fortuitous, or is there was a pattern there.

Based on that observation, consider the following transformation.

a (x^{2} + \frac{b}{a} x + \frac{b}{2 a}) + \frac{c}{a} - \frac{b}{2 a}

We are almost there, because this can be transformed into the following.

a {(x + \sqrt{\frac{b}{2 a}})}^{2} + \frac{c}{a} - \frac{b}{2 a}

And there we have it! Within the parethesis we have the desired binomial which, when squared, produces the reduced quadratic; which, when transformed backwards will transform to the original quadratic.

Summarising

Given the following general quadratic expression.

a x^{2} + b x + c

This can be transformed into the following expression containing a squared binomial expression.

a {(x + \sqrt{\frac{b}{2 a}})}^{2} + \frac{c}{a} - \frac{b}{2 a}

Limitations of the method

As can be seen from the previous expression, the method of completing the square invariably involves taking the square root in the term:

\sqrt{\frac{b}{2 a}}

From the form of this term the following can be deduced.

The parameter $a$ cannot be zero.
For a non-complex solution the term $\frac{b}{2 a}$ must evaluate to a value greater than or equal to zero. This implies that the parameters $a$ and $b$ must have the same sign; either both positive, or both negative.

Note: Complex solutions are permitted, but we will not be discussing that here.

Where to from here

Now that we have a method for reducing a general quadratic to an expression containing the square of a binimial expression we are in a position to develop this technique further to derive a formula for factorising a general quadratic.

Before you do that you should read the Note Concerning the Sign of a Square Root.

Mike Turnbull Completing the Square

Completing the Square

Introduction

Completing the Square

Summarising

Limitations of the method

Where to from here