# Completing the square

File:Completing the square.ogv

In elementary algebra, **completing the square** is a technique for converting a quadratic polynomial of the form

to the form

In this context, "constant" means not depending on *x*. The expression inside the parenthesis is of the form (*x* + constant). Thus

for some values of *h* and *k*.

Completing the square is used in

- solving quadratic equations,
- graphing quadratic functions,
- evaluating integrals in calculus, such as Gaussian integrals with a linear term in the exponent
- finding Laplace transforms.

In mathematics, completing the square is considered a basic algebraic operation, and is often applied without remark in any computation involving quadratic polynomials. Completing the square is also used to derive the quadratic formula.

## Overview

### Background

There is a simple formula in elementary algebra for computing the square of a binomial:

For example:

In any perfect square, the number *p* is always half the coefficient of *x*, and the constant term is equal to *p*^{2}.

### Basic example

Consider the following quadratic polynomial:

This quadratic is not a perfect square, since 28 is not the square of 5:

However, it is possible to write the original quadratic as the sum of this square and a constant:

This is called **completing the square**.

### General description

Given any monic quadratic

it is possible to form a square that has the same first two terms:

This square differs from the original quadratic only in the value of the constant term. Therefore, we can write

where *k* is a constant. This operation is known as **completing the square**.
For example:

### Non-monic case

Given a quadratic polynomial of the form

it is possible to factor out the coefficient *a*, and then complete the square for the resulting monic polynomial.

Example:

This allows us to write any quadratic polynomial in the form

### Formula

The result of completing the square may be written as a formula. For the general case:^{[1]}

Specifically, when *a*=1:

The matrix case looks very similar:

If is not symmetric the formulae for and have to be generalized to:

## Relation to the graph

{{#invoke:Multiple image|render}}
In analytic geometry, the graph of any quadratic function is a parabola in the *xy*-plane. Given a quadratic polynomial of the form

the numbers *h* and *k* may be interpreted as the Cartesian coordinates of the vertex of the parabola. That is, *h* is the *x*-coordinate of the axis of symmetry, and *k* is the minimum value (or maximum value, if *a* < 0) of the quadratic function.

One way to see this is to note that the graph of the function *ƒ*(*x*) = *x*^{2} is a parabola whose vertex is at the origin (0, 0). Therefore, the graph of the function *ƒ*(*x* − *h*) = (*x* − *h*)^{2} is a parabola shifted to the right by *h* whose vertex is at (*h*, 0), as shown in the top figure. In contrast, the graph of the function *ƒ*(*x*) + *k* = *x*^{2} + *k* is a parabola shifted upward by *k* whose vertex is at (0, *k*), as shown in the center figure. Combining both horizontal and vertical shifts yields *ƒ*(*x* − *h*) + *k* = (*x* − *h*)^{2} + *k* is a parabola shifted to the right by *h* and upward by *k* whose vertex is at (*h*, *k*), as shown in the bottom figure.

## Solving quadratic equations

Completing the square may be used to solve any quadratic equation. For example:

The first step is to complete the square:

Next we solve for the squared term:

Then either

and therefore

This can be applied to any quadratic equation. When the *x*^{2} has a coefficient other than 1, the first step is to divide out the equation by this coefficient: for an example see the non-monic case below.

### Irrational and complex roots

Unlike methods involving factoring the equation, which is only reliable if the roots are rational, completing the square will find the roots of a quadratic equation even when those roots are irrational or complex. For example, consider the equation

Completing the square gives

so

Then either

so

In terser language:

Equations with complex roots can be handled in the same way. For example:

### Non-monic case

For an equation involving a non-monic quadratic, the first step to solving them is to divide through by the coefficient of *x*^{2}. For example:

## Other applications

### Integration

Completing the square may be used to evaluate any integral of the form

using the basic integrals

For example, consider the integral

Completing the square in the denominator gives:

This can now be evaluated by using the substitution
*u* = *x* + 3, which yields

### Complex numbers

Consider the expression

where *z* and *b* are complex numbers, *z*^{*} and *b*^{*} are the complex conjugates of *z* and *b*, respectively, and *c* is a real number. Using the identity |*u*|^{2} = *uu*^{*} we can rewrite this as

which is clearly a real quantity. This is because

As another example, the expression

where *a*, *b*, *c*, *x*, and *y* are real numbers, with *a* > 0 and *b* > 0, may be expressed in terms of the square of the absolute value of a complex number. Define

Then

so

### Idempotent matrix

A matrix *M* is idempotent when *M* ^{2} = *M*. Idempotent matrices generalize the idempotent properties of 0 and 1. The completion of the square method of addressing the equation

shows that the idempotent 2 × 2 matrices are parametrized by a circle in the (*a,b*)-plane.

The matrix will be idempotent provided which, upon completing the square, becomes

In the (*a,b*)-plane, this is the equation of a circle with center (1/2, 0) and radius 1/2.

## Geometric perspective

Consider completing the square for the equation

Since *x*^{2} represents the area of a square with side of length *x*, and *bx* represents the area of a rectangle with sides *b* and *x*, the process of completing the square can be viewed as visual manipulation of rectangles.

Simple attempts to combine the *x*^{2} and the *bx* rectangles into a larger square result in a missing corner. The term (*b*/2)^{2} added to each side of the above equation is precisely the area of the missing corner, whence derives the terminology "completing the square". [1]

## A variation on the technique

As conventionally taught, completing the square consists of adding the third term, *v*^{ 2} to

to get a square. There are also cases in which one can add the middle term, either 2*uv* or −2*uv*, to

to get a square.

### Example: the sum of a positive number and its reciprocal

By writing

we show that the sum of a positive number *x* and its reciprocal is always greater than or equal to 2. The square of a real expression is always greater than or equal to zero, which gives the stated bound; and here we achieve 2 just when *x* is 1, causing the square to vanish.

### Example: factoring a simple quartic polynomial

Consider the problem of factoring the polynomial

This is

so the middle term is 2(*x*^{2})(18) = 36*x*^{2}. Thus we get

(the last line being added merely to follow the convention of decreasing degrees of terms).

## References

- ↑ {{#invoke:citation/CS1|citation
|CitationClass=book
}}, Section
*Formula for the Vertex of a Quadratic Function*, page 133–134, figure 2.4.8

- Algebra 1, Glencoe, ISBN 0-07-825083-8, pages 539–544
- Algebra 2, Saxon, ISBN 0-939798-62-X, pages 214–214, 241–242, 256–257, 398–401