Completing the square

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Animation depicting the process of completing the square. (Details, animated GIF version)

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

is converted to

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. Completing the square is also used to derive the quadratic formula.

Overview[edit]

Background[edit]

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 p2.

Basic example[edit]

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[edit]

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[edit]

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[edit]

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:

where has to be symmetric.

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

.

Relation to the graph[edit]

Graphs of quadratic functions shifted to the right by h = 0, 5, 10, and 15.
Graphs of quadratic functions shifted to the right by h = 0, 5, 10, and 15.
Graphs of quadratic functions shifted upward by k = 0, 5, 10, and 15.
Graphs of quadratic functions shifted upward by k = 0, 5, 10, and 15.
Graphs of quadratic functions shifted upward and to the right by 0, 5, 10, and 15.
Graphs of quadratic functions shifted upward and to the right by 0, 5, 10, and 15.

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) = x2 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) + kx2 + 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 (hk), as shown in the bottom figure.

Solving quadratic equations[edit]

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[edit]

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

so

In terser language:

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

Non-monic case[edit]

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:

Other applications[edit]

Integration[edit]

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[edit]

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[edit]

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 (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[edit]

Completing the square.svg

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 (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".

A variation on the technique[edit]

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[edit]

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[edit]

Consider the problem of factoring the polynomial

This is

so the middle term is 2(x2)(18) = 36x2. Thus we get

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

References[edit]

External links[edit]