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Completing the square
 
 
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In elementary algebra
Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic....
, completing the square is a technique for converting a quadratic polynomial
Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z....
 of the form

to the form



The expression inside the parenthesis is of the form x − constant. Thus one converts ax2 + bx + c to



and one must find h and k.

Completing the square is used in

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

e is a simple formula in elementary algebra
Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic....
 for computing the square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 of a binomial
Binomial

In elementary algebra, a binomial is a polynomial with two terms—the sum of two monomials—often bound by parenthesis or brackets when operated upon....
:

For example:

In any perfect square, the number p is always half the coefficient
Coefficient

In mathematics, a coefficient is a constant multiplication factor of a certain object. For example, in the expression 9x2, the coefficient of x2 is 9....
 of x, and then the constant term
Constant term

In mathematics, the constant term of a polynomial is the term of degree 0. For example, in the polynomialover the variable X, the constant term is 3....
 is equal to p2.

ider 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.

n any quadratic of the form

it it 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.






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Encyclopedia


In elementary algebra
Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic....
, completing the square is a technique for converting a quadratic polynomial
Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z....
 of the form

to the form



The expression inside the parenthesis is of the form x − constant. Thus one converts ax2 + bx + c to



and one must find h and k.

Completing the square is used in
  • solving quadratic equation
    Quadratic equation

    In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
    s,
  • graphing quadratic function
    Quadratic function

    A quadratic function, in mathematics, is a polynomial function of the form , where . The graph of a function of a quadratic function is a parabola whose major axis is parallel to the y-axis....
    s,
  • evaluating integral
    Integral

    Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
    s in calculus,
  • 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.

Overview


Background

There is a simple formula in elementary algebra
Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic....
 for computing the square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 of a binomial
Binomial

In elementary algebra, a binomial is a polynomial with two terms—the sum of two monomials—often bound by parenthesis or brackets when operated upon....
:

For example:

In any perfect square, the number p is always half the coefficient
Coefficient

In mathematics, a coefficient is a constant multiplication factor of a certain object. For example, in the expression 9x2, the coefficient of x2 is 9....
 of x, and then the constant term
Constant term

In mathematics, the constant term of a polynomial is the term of degree 0. For example, in the polynomialover the variable X, the constant term is 3....
 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 quadratic of the form

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

This allows us to write any quadratic polynomial in the form

Formulae

The result of completing the square may be written as a formula. Specifically,

For the general case:

Relation to the graph

In analytic geometry
Analytic geometry

Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
, the graph of any quadratic function
Quadratic function

A quadratic function, in mathematics, is a polynomial function of the form , where . The graph of a function of a quadratic function is a parabola whose major axis is parallel to the y-axis....
 is a parabola
Parabola

In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
 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
Maxima and minima

In mathematics, maxima and minima, known collectively as extrema, are the largest value or smallest value , that a function takes in a point either within a given neighbourhood or on the function domain in its entirety ....
 (or maximum value, if a < 0) of the quadratic function.

Solving quadratic equations

Completing the square may be used to solve any quadratic equation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
. Given a quadratic equation

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.) Indeed, this method can be used to derive the quadratic formula for the roots of a general quadratic polynomial (see Quadratic equation#Derivation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
).

Irrational roots

Unlike methods involving factoring
Factorization

In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplication together give the original....
 the constant term
Constant term

In mathematics, the constant term of a polynomial is the term of degree 0. For example, in the polynomialover the variable X, the constant term is 3....
, completing the square will find the roots of a quadratic equation even when those roots are irrational
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
. For example, consider the equation

Completing the square gives

so

Then either

and therefore

Complex roots

Completing the square can also find roots of a quadratic equation that are complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s:

Non-monic case

For an equation involving a non-monic quadratic, simply start by dividing through by the coefficient of x2. 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
Integration by substitution

In calculus, integration by substitution is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative....
 u = x + 3, which yields

Complex numbers

Consider the expression

where z and b are complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, z* and b* are the complex conjugate
Complex conjugate

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number...
s of z and b, respectively, and c is a real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
. 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
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
 of a complex number. Define

Then

so

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 (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 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(x2)(18) = 36x2. Thus we get

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

External links

  • Step By Step Solutions to Quadratic Equations by Completing the Square