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 functionƒ(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) + k = x2 + 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 (h, k), as shown in the bottom figure.
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
In terser language:
Equations with complex roots can be handled in the same way. For example:
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.
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 substitutionu = x + 3, which yields
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
A matrixM 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.
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
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
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
^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
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