Elementary Algebra Pq from Wikipedia, the Free Encyclopedia Contents

Elementary algebra pq From Wikipedia, the free encyclopedia Contents

1 Parent function 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 5.10 Text and image sources, contributors, and licenses ...... 42 5.10.1 Text ...... 42 5.10.2 Images ...... 43 5.10.3 Content license ...... 44 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 codomain Y, and elements of Y can be multiplied (for instance, Y could be some set 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 y z = yz · × −→ given by ·

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 ring (or ﬁeld), in which multiplication is well-deﬁned.

Y f g X • If is the set of real numbers R, then the pointwise product of , : → 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 convolution theorem states that the Fourier transform of a convolution is the pointwise product of Fourier transforms:

f g = f g F{ ∗ } F{ }·F{ }

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 addition and multiplication are defined in R, we can construct an algebraic 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

x2 x 1 = 0. The golden ratio is found as the solution of the quadratic equation − − 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 6 CHAPTER 3. QUADRATIC EQUATION

x q x s q s b c the form ( + )( + ), and one has to ﬁnd two numbers and that add up to and2 whose product is (this is sometimes called “Vieta’s rule”[3] and is related to Vieta’s formulas). As an example, x + 5x + 6 factors as (x + 3)(x + 2).The more general case where a does not equal 1 can require a considerable eﬀort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.[2]:207

3.2.2 Completing the square

Main article: Completing the square The process of completing the square makes use of the algebraic identity y 3

1 x −3 −2 −1 0 1 2 3 −1 y = x2−x−2 −2

−3

2 Figure 2. For the quadratic function = − − 2, the points where the graph crosses the -axis, = −1 and = 2, are the 2 y x x x x x solutions of the quadratic equation x − x − 2 = 0.

2 2 2 x + 2hx + h =(x + h) , 3.2. SOLVING THE QUADRATIC EQUATION 7

[2]:207 which represents a well-deﬁned algorithm2 that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, ax + bx + c = 0

1. Divide each side by a, the coeﬃcient of the squared term.

2. Rearrange the equation so that the constant term c/a is on the right side.

3. Add the square of one-half of b/a, the coeﬃcient of x, to both sides. This “completes the square”, converting the left side into a perfect square.

4. Write the left side as a square and simplify the right side if necessary.

5. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.

6. Solve the two linear equations.

2 We illustrate use of this algorithm by solving 2x + 4x − 4 = 0

2 1) x + 2x 2 = 0 − 2 2) x + 2x = 2 2 3) x + 2x + 1 = 2 + 1 2 4) (x + 1) = 3

5) x +1= √3 ± 6) x = 1 √3 − ± The plus-minus symbol "±" indicates that both x = −1 + √3 and x = −1 − √3 are solutions of the quadratic equation.[4]

3.2.3 Quadratic formula and its derivation

Main article: Quadratic formula

Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.[5] The mathematical proof will now be brieﬂy summarized.[6] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:

2 b b2 4ac x + = − . ( 2a) 4a2

Taking the square root of both sides, and isolating x, gives:

b √b2 4ac x = − ± − . 2a 2 ax bx Some sources,2 particularly older ones, use alternative parameterizations of the quadratic equation such as + 2 + c = 0 or ax − 2bx + c = 0 ,[7] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly diﬀerent forms for the solution, but are otherwise equivalent. A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or oﬀer insight into other areas of mathematics.