In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By subsequently isolating and taking the square root, a quadratic problem can be reduced to a linear problem.

The name completing the square comes from a geometrical picture in which represents an unknown length. Then the quantity represents the area of a square of side and the quantity represents the area of a pair of congruent rectangles with sides and . To this square and pair of rectangles, one more square is added, of side length . This crucial step completes a larger square of side length .

Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian clay tablets dating from 1800–1600 BCE. It was formalised and popularised by mathematician Al-Khwarizmi in his work Al-Jabr, and is still taught in elementary algebra courses today. It is also used for graphing quadratic functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials, for example in calculus evaluating Gaussian integrals with a linear term in the exponent, and finding Laplace transforms.

History

The technique of completing the square was known in the Old Babylonian Empire.

Muhammad ibn Musa Al-Khwarizmi, a famous polymath who wrote the early algebraic treatise Al-Jabr, used the technique of completing the square to solve quadratic equations.

Overview

Background

The formula in elementary algebra for computing the square of a binomial

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 is called completing the square.

General description

Given any monic

. This operation is known as completing the square

Non-monic case

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

This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms. The constant term at the end of the polynomial does not have to be included.

Formula

Scalar case

The result of completing the square may be written as a formula. In the general case, one has

In particular, when

and reorganizing the resulting expression, one gets the quadratic formula for the roots of the quadratic equation

Matrix case

The matrix has to be symmetric.

Relation to the graph

In analytic geometry, the graph of any quadratic function is a parabola in the xy the numbers h and k may be interpreted as the Cartesian coordinates of the vertex (or stationary point) of the parabola. That is, h is the x-coordinate of the axis of symmetry (i.e. the axis of symmetry has equation x = h), and k is the minimum value (or maximum value, if a0) of the quadratic function.

One way to see this is to note that the graph of the function is a parabola whose vertex is at the origin(0,0). Therefore, the graph of the function 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 is a parabola shifted upward by whose vertex is at , as shown in the center figure. Combining both horizontal and vertical shifts yields is a parabola shifted to the right by and upward by whose vertex is at , as shown in the bottom figure.

Solving quadratic equations

Completing the square may be used to solve any quadratic equation

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

Other applications

Integration

This can now be evaluated by using the substitution u=x+

Complex numbers

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*

where a, b, c, x, and y are real numbers, with a0 and b0, may be expressed in terms of the square of the absolute value

Idempotent matrix

A matrix M is idempotent when M2 = M shows that some idempotent 2×2 matrices are parametrized by a circle in the (a,b)-plane:

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

Geometric perspective

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

As conventionally taught, completing the square consists of adding the third term, v to get a square. There are also cases in which one can add the middle term, either 2uv or 2uv to get a square.

Example: the sum of a positive number and its reciprocal

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

so the middle term is 2(x2)(18)=36x2 (the last line being added merely to follow the convention of decreasing degrees of terms).

(Also known as Sophie Germain's identity).

Completing the cube

"Completing the square" consists to remark that the two first terms of a quadratic polynomial are also the first terms of the square of a linear polynomial, and to use this for expressing the quadratic polynomial as the sum of a square and a constant.

Completing the cube is a similar technique that allows to transform a cubic polynomial into a cubic polynomial without term of degree two.

More precisely, if

is a polynomial in its two first terms are the two first terms of the expanded form of

So, the change of variable

without term of degree two, which is called the depressed form of the original polynomial.

This transformation is generally the first step of the methods for solving the general cubic equation.

, which is called Tschirnhaus transformation.

References

• Algebra 1, Glencoe, , pages 539–544

• Algebra 2, Saxon, , pages 214–214, 241–242, 256–257, 398–401

External links