Elementary algebra pqtz From Wikipedia, the free encyclopedia Contents

1 Parent 1 1.1 See also ...... 1 1.2 External links ...... 1

2 Pointwise product 2 2.1 Formal definition ...... 2 2.2 Examples ...... 2 2.3 Algebraic application of pointwise products ...... 3 2.4 Generalization ...... 3 2.5 See also ...... 3

3 Quadratic equation 4 3.1 Examples and applications ...... 4 3.2 Solving the quadratic equation ...... 5 3.2.1 Factoring by inspection ...... 5 3.2.2 Completing the square ...... 6 3.2.3 Quadratic formula and its derivation ...... 7 3.2.4 Reduced quadratic equation ...... 8 3.2.5 Discriminant ...... 8 3.2.6 Geometric interpretation ...... 9 3.2.7 Quadratic factorization ...... 9 3.2.8 Graphing for real roots ...... 10 3.2.9 Avoiding loss of significance ...... 11 3.3 History ...... 11 3.4 Advanced topics ...... 13 3.4.1 Alternative methods of root calculation ...... 13 3.4.2 Generalization of quadratic equation ...... 16 3.5 See also ...... 18 3.6 References ...... 18 3.7 External links ...... 20

4 Quadratic formula 21 4.1 Derivation of the formula ...... 21

i ii CONTENTS

4.2 Geometrical Significance ...... 22 4.3 Historical Development ...... 23 4.4 Other derivations ...... 23 4.4.1 Alternate method of completing the square ...... 23 4.4.2 By substitution ...... 25 4.4.3 By using algebraic identities ...... 25 4.4.4 By Lagrange resolvents ...... 26 4.5 See also ...... 28 4.6 References ...... 28 4.7 External links ...... 29

5 Quartic function 30 5.1 History ...... 31 5.2 Examples ...... 31 5.3 Applications ...... 32 5.4 Inflection points and golden ratio ...... 32 5.5 Solving a quartic equation ...... 32 5.5.1 Nature of the roots ...... 32 5.5.2 General formula for roots ...... 33 5.5.3 Simpler cases ...... 34 5.5.4 Converting to a depressed quartic ...... 36 5.5.5 Ferrari’s solution ...... 37 5.5.6 Solving by factoring into quadratics ...... 38 5.5.7 Solving by Lagrange resolvent ...... 39 5.5.8 Solving with algebraic geometry ...... 40 5.6 See also ...... 40 5.7 References ...... 41 5.8 Further reading ...... 41 5.9 External links ...... 41

6 Rationalisation () 42 6.1 Rationalisation of a monomial square root and cube root ...... 42 6.2 Dealing with more square roots ...... 43 6.3 Generalisations ...... 44 6.4 See also ...... 44 6.5 References ...... 44

7 Solving quadratic equations with continued fractions 45 7.1 A simple example ...... 45 7.2 An algebraic explanation ...... 46 7.3 The general quadratic equation ...... 47 7.4 A general theorem ...... 47 CONTENTS iii

7.5 Complex coefficients ...... 48 7.6 See also ...... 49 7.7 References ...... 49

8 Transitive relation 50 8.1 Formal definition ...... 50 8.2 Examples ...... 50 8.3 Properties ...... 50 8.3.1 Closure properties ...... 51 8.3.2 Other properties ...... 51 8.3.3 Properties that require transitivity ...... 51 8.4 Counting transitive relations ...... 51 8.5 See also ...... 51 8.6 Sources ...... 52 8.6.1 References ...... 52 8.6.2 Bibliography ...... 52 8.7 External links ...... 52

9 Trinomial 53 9.1 Trinomial expressions ...... 53 9.2 Trinomial equation ...... 53 9.3 See also ...... 53 9.4 References ...... 53

10 Two-element Boolean algebra 54 10.1 Definition ...... 54 10.2 Some basic identities ...... 54 10.3 Metatheory ...... 55 10.4 Footnotes ...... 56 10.5 See also ...... 56 10.6 References ...... 56

11 Unary operation 57 11.1 Unary negative and positive ...... 57 11.2 Examples from programming languages ...... 57 11.2.1 C family of languages ...... 57 11.2.2 Unix Shell (Bash) ...... 58 11.2.3 Other languages ...... 58 11.3 See also ...... 58 11.4 References ...... 58

12 Unitary method 59 12.1 External links ...... 59 iv CONTENTS

13 Vieta’s formulas 60 13.1 The Laws ...... 60 13.1.1 Basic formulas ...... 60 13.1.2 Generalization to rings ...... 61 13.2 Example ...... 61 13.3 Proof ...... 61 13.4 History ...... 61 13.5 See also ...... 62 13.6 References ...... 62

14 Zero-product property 63 14.1 Algebraic context ...... 63 14.2 Examples ...... 63 14.3 Non-examples ...... 64 14.4 Application to finding roots of polynomials ...... 64 14.5 See also ...... 64 14.6 Notes ...... 65 14.7 References ...... 65 14.8 External links ...... 65 14.9 Text and sources, contributors, and licenses ...... 66 14.9.1 Text ...... 66 14.9.2 Images ...... 68 14.9.3 Content license ...... 69 Chapter 1

Parent function

In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions having the general form

2 y = ax + bx + c,

the simplest function is

2 y = x

This is therefore the parent function of the family of quadratic equations.

For linear and quadratic functions, the graph of any function can be obtained from the2 graph of the parent function by y x x simple translations2 and stretches parallel to the axes. For example, the graph of = − 4 + 7 can be obtained from y x the graph of = by translating +22 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2) . For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α A B R along2 the2 positive2 X axis (where tan(α) = ⁄ ), then stretching it parallel to the Y axis using a stretch factor , where R = A + B . This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities). The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but n for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as x , or, to 2 3 simplify further, x when n is even and x for odd n. Turning points may be established by differentiation to provide more detail of the graph.

1.1 See also

• Curve sketching

1.2 External links

• Video explanation at VirtualNerd.com

1 Chapter 2

Pointwise product

For entrywise product, see Matrix multiplication#Hadamard product.

The pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain. If f and g are both functions with domain X and Y, and elements of Y can be multiplied (for instance, Y could be some of numbers), then the pointwise product of f and g is another function from X to Y which maps x ∈ X to f(x)g(x).

2.1 Formal definition

Let X and Y be sets, and let multiplication be defined in Y—that is, for each y and z in Y let the product

· : Y × Y −→ Y given by y · z = yz

be well-defined. Let f and g be functions f, g : X → Y. Then the pointwise product (f ⋅ g): X → Y is defined by

(f · g)(x)= f(x) · g(x) for each x in X. In the same manner in which the binary operator ⋅ is omitted from products, we say that f ⋅ g = fg.

2.2 Examples

The most common case of the pointwise product of two functions is when the codomain is a (or field), in which multiplication is well-defined.

• If Y is the set of real numbers R, then the pointwise product of f, g : X → R is just normal multiplication of the images. For example, if we have f(x) = 2x and g(x) = x + 1 then

2 (fg)(x)= f(x)g(x) = 2x(x + 1) = 2x + 2x for every real number x in R.

• The theorem states that the of a convolution is the pointwise product of Fourier transforms:

F{f ∗ g} = F{f}·F{g}

2 2.3. ALGEBRAIC APPLICATION OF POINTWISE PRODUCTS 3

2.3 Algebraic application of pointwise products

Let X be a set and let R be a ring. Since and multiplication are defined in R, we can construct an alge- braic structure known as an algebra out of the functions from X to R by defining addition, multiplication, and scalar multiplication of functions to be done pointwise. X X If R denotes the set of functions from X to R, then we say that if f, g are elements of R , then f + g, fg, and rf, the last of which is defined by

(rf)(x)= rf(x)

X for all r in R, are all elements of R .

2.4 Generalization

If both f and g have as their domain all possible assignments of a set of discrete variables, then their pointwise product is a function whose domain is constructed by all possible assignments of the union of both sets. The value of each assignment is calculated as the product of the values of both functions given to each one the subset of the assignment that is in its domain. For example, given the function f1() for the boolean variables p and q, and f2() for the boolean variables q and r, both with the range in R, the pointwise product of f1() and f2() is shown in the next table:

2.5 See also

• Pointwise Chapter 3

Quadratic equation

This article is about single-variable quadratic equations and their solutions. For more general information about the single-variable case, see Quadratic function. For the case of more than one variable, see Quadratic form. In elementary algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the

The quadratic formula for the roots of the general quadratic equation form

2 ax + bx + c = 0 where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.[1] Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two. Quadratic equations can be solved by a process known in American English as factoring and in other varieties of English as factorising, by completing the square, by using the quadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.

3.1 Examples and applications

2 The golden ratio is found as the solution of the quadratic equation x − x − 1 = 0. The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.

4 3.2. SOLVING THE QUADRATIC EQUATION 5

Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation. Descartes’ theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation. The equation given by Fuss’ theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles’ centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle’s center and the center of the excircle of an ex-tangential quadrilateral.

3.2 Solving the quadratic equation

2 Figure 1. Plots of quadratic function y = ax + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0)

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

3.2.1 Factoring by inspection 2 It may be possible to express a quadratic equation ax + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the “Zero Factor Property” states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.

For most students, factoring by inspection is the first method of solving2 quadratic equations to which they are exposed.[2]:202–207 If one is given a quadratic equation in the form x + bx + c = 0, the sought factorization has