Completing the square


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

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

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.



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 exampleEdit

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 descriptionEdit

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 caseEdit

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.



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



This allows the writing of any quadratic polynomial in the form



Scalar caseEdit

The result of completing the square may be written as a formula. In the general case, one has[1]




In particular, when a = 1, one has




By solving the equation   in terms of   and reorganizing the resulting expression, one gets the quadratic formula for the roots of the quadratic equation:


Matrix caseEdit

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 graphEdit

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 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 (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 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 equationsEdit

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 rootsEdit

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




Then either


In terser language:




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


Non-monic caseEdit

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 applicationsEdit


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 numbersEdit

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






Idempotent matrixEdit

A matrix M is idempotent when M2 = 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 perspectiveEdit


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 techniqueEdit

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 reciprocalEdit

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 polynomialEdit

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).

The same argument shows that   is always factorizable as


(Also known as Sophie Germain's identity).


  1. ^ Narasimhan, Revathi (2008). Precalculus: Building Concepts and Connections. Cengage Learning. pp. 133–134. ISBN 978-0-618-41301-0., 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

External linksEdit