NationMaster - Encyclopedia: Completing the square

Completing the square is an algebra technique, also used in many types of calculus. The essential objective is to reduce all instances of the variable in an equation or expression to be the same order. This makes the equation easier to solve in many contexts. Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... An expression in the very basic sense is the noun form of the verb express. ... The degree of a term of a polynomial in one variable is the exponent on the variable in that term; the degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ... An equation is a mathematical statement, in symbols, that two things are the same. ...

Very simply, given a problem of the form:

x 2 + b x + c = 0

We may transform the equation such that we no longer have both "x" and "x squared" with a method called "completing the square". To do this we take one half of the coefficient of "x", square it, and add it to both sides of the equation, like so:

$begin{matrix} x^2 + bx +c &=& 0 x^2 + bx &=& -c x^2 + bx + (frac{b}{2})^2 &=& -c+ (frac{b}{2})^2 (x + frac{b}{2})^2 &=& (frac{b}{2})^2 - c (x + frac{b}{2}) &=& pmsqrt{ (frac{b}{2})^2 - c} x &=& - frac{b}{2}pmsqrt{ (frac{b}{2})^2 - c} end{matrix}$

More specifically, Completing the square is a technique of elementary algebra wherein an expression Elementary algebra is the most basic form of algebra taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. ...

x 2 + b x

is replaced by one of the form

(x + c) 2 + d

Specifically, we have:

$ax^2 + bx + c = aleft(x^2 + frac{bx}{a}right) +c$

$= aleft(x^2 + frac{bx}{a} + left(frac{b^2}{4a^2} - frac{b^2}{4a^2}right)right) + c$

$= aleft(x^2+2frac{bx}{2a}+left(frac{b}{2a}right)^2right)-frac{b^2}{4a} +c$

$= aleft(x+frac{b}{2a}right)^2-frac{b^2}{4a} + c.$

Completing the square reduces any problem involving a quadratic polynomial to one involving a square quadratic polynomial and a constant. It should be noted that the coefficients a, b and c above can be expressions in their own right, containing variables other than x. f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...

See also: quadratic equation Graph of a quadratic function: y = x2 â€“ x â€“ 2 = (x + 1)(x â€“ 2) The x-coordinates of the points where the graph crosses the x-axis, x = â€“1 and x = 2, are the roots of the quadratic equation: x2 â€“ x â€“ 2 = 0 In mathematics, a quadratic equation is a polynomial...

1 Formula for completing the square
2 Example
3 Complex versions of completing the square
4 External link

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Formula for completing the square

For $a x 2 + b x + c$ ,

$(xsqrt{a}+frac{b}{2sqrt{a}})^2+c-(frac{b}{2sqrt{a}})^2$

There is also an easy way to remember it: After completing the square,

$(e x + f) 2 + g$ ,

$e=sqrt{a}$

$f=frac{b}{2sqrt{a}}$

$g = c - f 2$

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Example

A very simple example is:

$begin{matrix} x^2 + 6x - 16 &=& 0 x^2 + 6x &=& 16 x^2 + 6x + (frac{6}{2})^2 &=& 16 + (frac{6}{2})^2 x^2 + 6x + 9 &=& 16 + 9 (x + 3)^2 &=& 25 (x + 3) &=& sqrt{25} x + 3 &=& pm5 x &=& pm5 - 3 x &=& - 8 , 2 end{matrix}$

Another fairly simple example is:

x 2 + 4 x = x 2 + 4 x + 4 - 4 = (x + 2) 2 - 4

Now, consider the problem of finding this antiderivative: In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

$intfrac{dx}{9x^2-90x+241}.$

The denominator is In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...

9 x 2 - 90 x + 241 = 9(x 2 - 10 x) + 241.

Adding (10/2)² = 25 to x² - 10x gives a perfect square x² - 10x + 25 = (x - 5)². So we get

9(x 2 - 10 x) + 241 = 9(x 2 - 10 x + 25) + 241 - 9(25) = 9(x - 5) 2 + 16.

Let our integral be In calculus, the integral of a function is a generalization of area, mass, volume and total. ...

$intfrac{dx}{9x^2-90x+241}=frac{1}{9}intfrac{dx}{(x-5)^2+(4/3)^2}=frac{1}{9}cdotfrac{3}{4}arctanfrac{3(x-5)}{4}+C.$

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Complex versions of completing the square

Consider the expression

| z | 2 - b * z - b z * + c

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. This can be re-expressed as The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...

| z - b | 2 - | b | 2 + c

which is clearly a real quantity. This is because

$begin{matrix} |z-b|^2 &=& (z-b)(z-b)^* &=& (z-b)(z^*-b^*) &=& zz^* - zb^* - bz^* + bb^* &=& |z|^2 - zb^* - bz^* + |b|^2 end{matrix}$

Also, the expression

a x 2 + b y 2 + c

where $a$ , $x$ , $b$ , $y$ , and $c$ 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. Defining

$z equiv sqrt{a} x + i sqrt{b} y$ ,

we then have

$begin{matrix} |z|^2 &=& z z^* &=& (sqrt{a} x + i sqrt{b} y)(sqrt{a} x - i sqrt{b} y) &=& ax^2 - isqrt{a}sqrt{b}xy + isqrt{b}sqrt{a}yx - i^2by^2 &=& ax^2 + by^2 end{matrix}$

a x 2 + b y 2 + c = | z | 2 + c

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External link

Completing the square on PlanetMath

Category: Algebra PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Results from FactBites:

Completing the Square: Circle Equations (456 words)
	The technique of completing the square is used to turn a quadratic into a squared binomial, plus some loose numbers:
	, completing the square can be necessary for finding the center and radius of a circle, once you've been given the equation.
	Whatever is multiplied on the squared terms (it'll always be the same number!), divide it off from every term.

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Formula for completing the square GA_googleFillSlot("encyclopedia_square");

Example

Complex versions of completing the square

External link

Formula for completing the square