Elementary algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search
The quadratic formula, which is the solution to the quadratic equation ax^2+bx+c=0 . Here the symbols a,b,c,x all are variables that represent numbers.
Two-dimensional plot (magenta curve) of the algebraic equation y = x^2 - x - 2

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers,[1] algebra introduces quantities without fixed values, known as variables.[2] This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic equations.

Algebraic notation[edit]

Main article: Mathematical notation

Algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology. For example, the expression 3x^2 - 2xy + c has the following components:

Algebraic equation notation.svg
1 : Exponent (power), 2 : Coefficient, 3 : term, 4 : operator, 5 : constant, x, y : variables

A coefficient is a numerical value which multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.[3] Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. a, b, c) are typically used to represent constants, and those toward the end of the alphabet (e.g. x, y and z) are used to represent variables.[4] They are usually written in italics.[5]

Algebraic operations work in the same way as arithmetic operations,[6] such as addition, subtraction, multiplication, division and exponentiation.[7] and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 \times x^2 is written as 3x^2, and 2 \times x \times y may be written 2xy.[8]

Usually terms with the highest power (exponent), are written on the left, for example, x^2 is written to the left of x. When a coefficient is one, it is usually omitted (e.g. 1x^2 is written x^2).[9] Likewise when the exponent (power) is one, (e.g. 3x^1 is written 3x).[10] When the exponent is zero, the result is always 1 (e.g. x^0 is always rewritten to 1).[11] However 0^0, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

Alternative notation[edit]

Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. For example, exponents are usually formatted using superscripts, e.g. x^2. In plain text, and in the TeX mark-up language, the caret symbol "^" represents exponents, so x^2 is written as "x^2".[12][13] In programming languages such as Ada,[14] Fortran,[15] Perl,[16] Python [17] and Ruby,[18] a double asterisk is used, so x^2 is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,[19] and it must be explicitly used, for example, 3x is written "3*x".

Concepts[edit]

Variables[edit]

Example of variables showing the relationship between a circle's diameter and its circumference. For any circle, its circumference c, divided by its diameter d, is equal to the constant pi, \pi (approximately 3.14).

Elementary algebra builds on and extends arithmetic[20] by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

  1. Variables may represent numbers whose values are not yet known. For example, if the temperature today, T, is 20 degrees higher than the temperature yesterday, Y, then the problem can be described algebraically as T = Y + 20.[21]
  2. Variables allow one to describe general problems,[22] without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to 60 \times 5 = 300 seconds. A more general (algebraic) description may state that the number of seconds, s = 60 \times m, where m is the number of minutes.
  3. Variables allow one to describe mathematical relationships between quantities that may vary.[23] For example, the relationship between the circumference, c, and diameter, d, of a circle is described by \pi = c /d.
  4. Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as (a + b) = (b + a).[24]

Evaluating expressions[edit]

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,

  • Added terms are simplified using coefficients. For example x + x + x can be simplified as 3x (where 3 is the coefficient).
  • Multiplied terms are simplified using exponents. For example x \times x \times x is represented as x^3
  • Like terms are added together,[25] for example, 2x^2 + 3ab - x^2 + ab is written as x^2 + 4ab, because the terms containing x^2 are added together, and, the terms containing ab are added together.
  • Brackets can be "multiplied out", using distributivity. For example, x (2x + 3) can be written as (x \times 2x) + (x \times 3) which can be written as 2x^2 + 3x
  • Expressions can be factored. For example, 6x^5 + 3x^2, by dividing both terms by 3x^2 can be written as 3x^2 (2x^3 + 1)

Equations[edit]

Animation illustrating Pythagoras' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.
Main article: Equation

An equation states that two expressions are equal using the symbol for equality, = (the equals sign).[26] One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:[27]

c^2 = a^2 + b^2

This equation states that c^2, representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by a and b.

An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as a + b = b + a); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. x^2 - 1 = 8 is true only for x = 3 and x = -3. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.

Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are:  a > b where  > represents 'greater than', and  a < b where  < represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.

Properties of equality[edit]

By definition, equality is an equivalence relation, meaning it has the properties (a) reflexive (i.e. b = b), (b) symmetric (i.e. if a = b then b = a) (c) transitive (i.e. if a = b and b = c then a = c).[28] It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:

  • if a = b and c = d then a + c = b + d and ac = bd;
  • if a = b then a + c = b + c;
  • more generally, for any function f, a=b implies f(a) = f(b).

Properties of inequality[edit]

The relations less than  < and greater than  > have the property of transitivity:[29]

  • If   a < b   and   b < c   then   a < c;
  • If   a < b   and   c < d   then   a + c < b + d;
  • If   a < b   and   c > 0   then   ac < bc;
  • If   a < b   and   c < 0   then   bc < ac.

Note that by reversing the equation, we can swap  < and  > ,[30] for example:

  • a < b is equivalent to b > a

Substitution[edit]

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for a in the expression a*5 makes a new expression 3*5 with meaning 15. Substituting the terms of a statement makes a new statement. When the original statement is true independent of the values of the terms, the statement created by substitutions is also true. Hence definitions can be made in symbolic terms and interpreted through substitution: if a^2:=a*a, where := means "is defined to equal", substituting 3 for a informs the reader of this statement that 3^2 means 3*3=9. Often it's not known whether the statement is true independent of the values of the terms, and substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x+1=0, if x is substituted with 1, this imples 1+1=2=0, which is false, which implies that if x+1=0 then x can't be 1.

If x and y integers, rationals, or real numbers, then xy=0 implies x=0 or y=0. Suppose abc=0. Then, substituting a for x and bc for y, we learn a=0 or bc=0. Then we can substitute again, letting x=b and y=c, to show that if bc=0 then b=0 or c=0. Therefore if abc=0, then a=0 or (b=0 or c=0), so abc=0 implies a=0 or b=0 or c=0.

Consider if the original fact were stated as "ab=0 implies a=0 or b=0." Then when we say "suppose abc=0," we have a conflict of terms when we substitute. Yet the above logic is still valid to show that if abc=0 then a=0 or b=0 or c=0 if instead of letting a=a and b=bc we substitute a for a and b for bc (and with bc=0, substituting b for a and c for b). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression a into the a term of the original equation, the a substituted does not refer to the a in the statement "ab=0 implies a=0 or b=0."

Solving algebraic equations[edit]

A typical algebra problem.

The following sections lay out examples of some of the types of algebraic equations that may be encountered.

Linear equations with one variable[edit]

Main article: Linear equation

Linear equations are so-called, because when they are plotted, they describe a straight line (hence linear). The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:

Problem in words: If you double my son's age and add 4, the resulting answer is 12. How old is my son?
Equivalent equation: 2x + 4 = 12 where x represent my son's age

To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.[31] This problem and its solution are as follows:

1. Equation to solve: 2x + 4 = 12
2. Subtract 4 from both sides: 2x + 4 - 4 = 12 - 4
3. This simplifies to: 2x = 8
4. Divide both sides by 2: \frac{2x}{2} = \frac{8}{2}
5. Simplifies to the solution: x = 4

The general form of a linear equation with one variable, can be written as: ax+b=c\,

Following the same procedure (i.e. subtract b from both sides, and then divide by a), the general solution is given by x=\frac{c-b}{a}

Linear equations with two variables[edit]

Solving two linear equations with a unique solution at the point that they intersect.

A linear equation with two variables has many (i.e. an infinite number of) solutions.[32] For example:

Problem in words: I am 22 years older than my son. How old are we?
Equivalent equation: y = x + 22 where y is my age, x is my son's age.

This can not be worked out by itself. If I told you my son's age, then there would no longer be two unknowns (variables), and the problem becomes a linear equation with just one variable, that can be solved as described above.

To solve a linear equation with two variables (unknowns), requires two related equations. For example, if I also revealed that:

Problem in words: In 10 years time, I will be twice as old as my son.
Equivalent equation: y + 10 = 2 \times (x + 10)
Subtract 10 from both sides: y = 2 \times (x + 10) - 10
Multiple out brackets: y = 2x + 20 - 10
Simplify: y = 2x + 10

Now there are two related linear equations, each with two unknowns, which lets us produce a linear equation with just one variable, by subtracting one from the other (called the elimination method):[33]

Second equation y = 2x + 10
First equation y = x + 22
Subtract the first equation from
the second in order to remove y
(y - y) = (2x - x) +10 - 22
Simplify 0 = x - 12
Add 12 to both sides 12 = x
Rearrange x = 12

In other words, my son is aged 12, and as I am 22 years older, I must be 34. In 10 years time, my son will 22, and I will be twice his age, 44. This problem is illustrated on the associated plot of the equations.

For other ways to solve this kind of equations, see below, System of linear equations.

Quadratic equations[edit]

Main article: Quadratic equation
Quadratic equation plot of y = x^2 + 3x - 10 showing its roots at x = -5 and x = 2, and that the quadratic can be rewritten as y = (x + 5)(x - 2)

A quadratic equation is one which includes a term with an exponent of 2, for example, x^2,[34] and no term with higher exponent. The name derives from the Latin quadrus, meaning square.[35] In general, a quadratic equation can be expressed in the form ax^2 + bx + c = 0,[36] where a is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term ax^2, which is known as the quadratic term. Hence a \neq 0, and so we may divide by a and rearrange the equation into the standard form

x^2 + px + q = 0 \,

where p = b/a and q = c/a. Solving this, by a process known as completing the square, leads to the quadratic formula

x=\frac{-b \pm \sqrt {b^2-4ac}}{2a},

where the symbol "±" indicates that both

 x=\frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and}\quad x=\frac{-b - \sqrt {b^2-4ac}}{2a}

are solutions of the quadratic equation.

Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:

x^{2} + 3x - 10 = 0. \,

Which is the same thing as

(x + 5)(x - 2) = 0. \,

It follows from the zero-product property that either x = 2 or x = -5 are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,

x^{2} + 1 = 0 \,

has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:

(x + 1)^2 = 0. \,

For this equation, −1 is a root of multiplicity 2. This means −1 appears two times.

Exponential and logarithmic equations[edit]

Main article: Logarithm
Graph showing a logarithm curves, which crosses the x-axis where x is 1 and extend towards minus infinity along the y-axis.
The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.

An exponential equation is one which has the form a^x = b for a > 0,[37] which has solution

X = \log_a b = \frac{\ln b}{\ln a}

when b > 0. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if

3 \cdot 2^{x - 1} + 1 = 10

then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

2^{x - 1} = 3\,

whence

x - 1 = \log_2 3\,

or

x = \log_2 3 + 1.\,

A logarithmic equation is an equation of the form log_a(x) = b for a > 0, which has solution

X = a^b.\,

For example, if

4\log_5(x - 3) - 2 = 6\,

then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get

\log_5(x - 3) = 2\,

whence

x - 3 = 5^2 = 25\,

from which we obtain

x = 28.\,

Radical equations[edit]

Radical equation showing two ways to represent the same expression

A radical equation is one that includes a radical sign, which includes square roots, \sqrt{x}, cube roots, \sqrt[3]{x}, and nth roots, \sqrt[n]{x}. Recall that an nth root can be rewritten in exponential format, so that \sqrt[n]{x} is equivalent to x^{\frac{1}{n}}. Combined with regular exponents (powers), then \sqrt[2]{x^3} (the square root of x cubed), can be rewritten as x^{\frac{3}{2}}.[38] So a common form of a radical equation is a = \sqrt[n]{x^m} (equivalent to a = x^\frac{m}{n}) where m and n are integers. It has solution:

m is odd m is even
and a \ge 0
x = \sqrt[m]{a^n}

or

x = \left(\sqrt[m]a\right)^n
x = \pm \sqrt[m]{a^n}

or

x = \pm \left(\sqrt[m]a\right)^n

For example, if:

(x + 5)^{2/3} = 4,\,

then

\begin{align}
x + 5 & = \pm (\sqrt{4})^3\\
x + 5 & = \pm 8\\
x &  = -5 \pm 8\\
x & = 3,-13
\end{align}.

System of linear equations[edit]

There are different methods to solve a system of linear equations with two variables.

Elimination method[edit]

The solution set for the equations x - y = -1 and 3x + y = 9 is the single point (2, 3).

An example of solving a system of linear equations is by using the elimination method:

\begin{cases}4x + 2y&= 14 \\
2x - y&= 1.\end{cases} \,

Multiplying the terms in the second equation by 2:

4x + 2y = 14 \,
4x - 2y = 2. \,

Adding the two equations together to get:

8x = 16 \,

which simplifies to

x = 2. \,

Since the fact that x = 2 is known, it is then possible to deduce that y = 3 by either of the original two equations (by using 2 instead of x ) The full solution to this problem is then

\begin{cases} x = 2 \\ y = 3. \end{cases}\,

Note that this is not the only way to solve this specific system; y could have been solved before x.

Substitution method[edit]

Another way of solving the same system of linear equations is by substitution.

\begin{cases}4x + 2y &= 14
\\ 2x - y &= 1.\end{cases} \,

An equivalent for y can be deduced by using one of the two equations. Using the second equation:

2x - y = 1 \,

Subtracting 2x from each side of the equation:

\begin{align}2x - 2x - y & = 1 - 2x  \\
- y & = 1 - 2x
\end{align}

and multiplying by −1:

 y = 2x - 1. \,

Using this y value in the first equation in the original system:

\begin{align}4x + 2(2x - 1) &= 14\\
4x + 4x - 2 &= 14 \\
8x - 2 &= 14 \end{align}

Adding 2 on each side of the equation:

\begin{align}8x - 2 + 2 &= 14 + 2 \\
8x &= 16 \end{align}

which simplifies to

x = 2 \,

Using this value in one of the equations, the same solution as in the previous method is obtained.

\begin{cases} x = 2 \\ y = 3. \end{cases}\,

Note that this is not the only way to solve this specific system; in this case as well, y could have been solved before x.

Other types of systems of linear equations[edit]

Inconsistent systems[edit]

The equations 3x + 2y = 6 and 3x + 2y = 12 are parallel and cannot intersect, and is unsolvable.

In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is

\begin{cases}\begin{align} x + y &= 1 \\
0x + 0y &= 2\,. \end{align} \end{cases}

As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, let us consider the system

\begin{cases}\begin{align}4x + 2y &= 12 \\
-2x - y &= -4\,. \end{align}\end{cases}

Multiplying by 2 both sides of the second equation, and adding it to the first one results in

0x+0y = 4 \,,

which has clearly no solution.

Undetermined systems[edit]

Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution.

There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for x and y) For example:

\begin{cases}\begin{align}4x + 2y & = 12 \\
-2x - y & = -6 \end{align}\end{cases}\,

Isolating y in the second equation:

y = -2x + 6 \,

And using this value in the first equation in the system:

\begin{align}4x + 2(-2x + 6) = 12 \\
4x - 4x + 12 = 12 \\
12 = 12 \end{align}

The equality is true, but it does not provide a value for x. Indeed, one can easily verify (by just filling in some values of x) that for any x there is a solution as long as y = -2x + 6. There is an infinite number of solutions for this system.

Over- and underdetermined systems[edit]

Systems with more variables than the number of linear equations do not have a unique solution. An example of such a system is

\begin{cases}\begin{align}x + 2y & = 10\\
y - z  & = 2 \end{align}\end{cases}

Such a system is called underdetermined; when trying to solve it, one is led to express some variables as functions of the other ones, but cannot express all solutions numerically.

A system with a greater number of equations than variables, in which necessarily some equations are linear combinations of the others if any solution exists, is called overdetermined.

See also[edit]

References[edit]

  1. ^ H.E. Slaught and N.J. Lennes, Elementary algebra, Publ. Allyn and Bacon, 1915, page 1 (republished by Forgotten Books)
  2. ^ Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, page 2
  3. ^ Richard N. Aufmann, Joanne Lockwood, Introductory Algebra: An Applied Approach, Publisher Cengage Learning, 2010, ISBN 1439046042, 9781439046043, page 78
  4. ^ William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71
  5. ^ James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425, 9780387985428, 221 pages, [James E. Gentle page 183]
  6. ^ Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, page 7
  7. ^ Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: Cengage Learning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, page 6
  8. ^ Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in Mathematics Matters Secondary 1 Express Textbook, Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, page 68
  9. ^ David Alan Herzog, Teach Yourself Visually Algebra, Publisher John Wiley & Sons, 2008, ISBN 0470185597, 9780470185599, 304 pages, page 72
  10. ^ John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 31
  11. ^ Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, Publisher Cengage Learning, 2010, ISBN 0538733543, 9780538733540, 803 pages, page 222
  12. ^ Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153, 9789380298153, page 212
  13. ^ George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, ISBN 0817641327, 9780817641320, page 17
  14. ^ S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual, Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352, page 13
  15. ^ C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, ISBN 812240670X, 9788122406702, page 20
  16. ^ Randal Schwartz, Brian Foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140, 9781449313142, page 24
  17. ^ Matthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, ISBN 1598631586, 9781598631586, page 46
  18. ^ Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, ISBN 1593271484, 9781593271480, page 72
  19. ^ William P. Berlinghoff, Fernando Q. Gouvêa, Math through the Ages: A Gentle History for Teachers and Others, Publisher MAA, 2004, ISBN 0883857367, 9780883857366, page 75
  20. ^ Thomas Sonnabend, Mathematics for Teachers: An Interactive Approach for Grades K-8, Publisher: Cengage Learning, 2009, ISBN 0495561665, 9780495561668, 759 pages, page xvii
  21. ^ Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, page 48
  22. ^ Lawrence S. Leff, College Algebra: Barron's Ez-101 Study Keys, Publisher: Barron's Educational Series, 2005, ISBN 0764129147, 9780764129148, 230 pages, page 2
  23. ^ Ron Larson, Kimberly Nolting, Elementary Algebra, Publisher: Cengage Learning, 2009, ISBN 0547102275, 9780547102276, 622 pages, page 210
  24. ^ Charles P. McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, ISBN 0840064217, 9780840064219, 571 pages, page 49
  25. ^ Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, ISBN 1419552880, 9781419552885, 288 pages, page 51
  26. ^ Mark Clark, Cynthia Anfinson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learning, 2011, ISBN 0534419380, 9780534419387, 793 pages, page 134
  27. ^ Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, ISBN 1111567689, 9781111567682, 1163 pages, page 493
  28. ^ Douglas Downing, Algebra the Easy Way, Publisher Barron's Educational Series, 2003, ISBN 0764119729, 9780764119729, 392 pages, page 20
  29. ^ Ron Larson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, ISBN 0618753524, 9780618753529, 857 pages, page 96
  30. ^ Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, ISBN 019914768X, 9780199147687, 144 pages, page 50
  31. ^ Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. p. 72. ISBN 0-471-50636-2. 
  32. ^ Sinha, The Pearson Guide to Quantitative Aptitude for CAT 2/ePublisher: Pearson Education India, 2010, ISBN 8131723666, 9788131723661, 599 pages, page 195
  33. ^ Cynthia Y. Young, Precalculus, Publisher John Wiley & Sons, 2010, ISBN 0471756849, 9780471756842, 1175 pages, page 699
  34. ^ Mary Jane Sterling, Algebra II For Dummies, Publisher: John Wiley & Sons, 2006, ISBN 0471775819, 9780471775812, 384 pages, page 37
  35. ^ John T. Irwin, The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story, Publisher JHU Press, 1996, ISBN 0801854660, 9780801854668, 512 pages, page 372
  36. ^ Sharma/khattar, The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E, Publisher Pearson Education India, 2010, ISBN 8131723631, 9788131723630, 1248 pages, page 621
  37. ^ Aven Choo, LMAN OL Additional Maths Revision Guide 3, Publisher Pearson Education South Asia, 2007, ISBN 9810600011, 9789810600013, page 105
  38. ^ John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 525
  39. ^ Euler's Elements of Algebra
  40. ^ Elements of algebra – Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange – Google Books

External links[edit]