Completing the square


In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
to the form
for some values of h and k.
Completing the square is used in
In mathematics, completing the square is often applied in any computation involving quadratic polynomials.

Overview

Background

The formula in elementary algebra for computing the square of a binomial is:
For example:
In any perfect square, the coefficient of x is twice the number p, and the constant term is equal to p2.

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. 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

Scalar case

The result of completing the square may be written as a formula. For the general case:
Specifically, when a = 1:

Matrix case

The matrix case looks very similar:
where has to be symmetric.
If is not symmetric the formulae for and have
to be generalized to:

Relation to the graph

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 of the quadratic function.
One way to see this is to note that the graph of the function ƒ = x2 is a parabola whose vertex is at the origin . Therefore, the graph of the function ƒ = 2 is a parabola shifted to the right by h whose vertex is at, as shown in the top figure. In contrast, the graph of the function ƒ + k = x2 + k is a parabola shifted upward by k whose vertex is at, as shown in the center figure. Combining both horizontal and vertical shifts yields ƒ + k = 2 + k is a parabola shifted to the right by h and upward by k whose vertex is at, 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 x2 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 reliable only 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
In terser language:
so
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 x2. For example:
Applying this procedure to the general form of a quadratic equation leads to the quadratic formula.

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 some idempotent 2 × 2 matrices are parametrized by a circle in the -plane:
The matrix will be idempotent provided which, upon completing the square, becomes
In the -plane, this is the equation of a circle with center and radius 1/2.

Geometric perspective

Consider completing the square for the equation
Since x2 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 x2 and the bx rectangles into a larger square result in a missing corner. The term 2 added to each side of the above equation is precisely the area of the missing corner, whence derives the terminology "completing the square".

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 2uv or −2uv, 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 = 36x2. Thus we get
.
The same argument shows that is always factorizable as
.