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Algebra

The algebra section of QuickMath allows you to manipulate mathematical expressions in all sorts of useful ways. At the moment, QuickMath can expand, factor or simplify virtually any expression, cancel common factors within fractions, split fractions up into smaller ('partial') fractions and join two or more fractions together into a single fraction. More specialized commands are on the way.

What is algebra?

The term 'algebra' is used for many things in mathematics, but in this section we'll just be talking about the sort of algebra you come across at high-school.

Algebra is the branch of elementary mathematics which uses symbols to stand for unknown quantities. In a more basic sense, it consists of solving equations or manipulating expressions which contain symbols (usually letters, like x, y or z) as well as numbers and functions. Although solving equations is really a part of algebra, it is such a big area that it has its own section in QuickMath.

This part of QuickMath deals only with algebraic expressions. These are mathematical statements which contain letters, numbers and functions, but no equals signs. Here are a few examples of simple algebraic expressions :

x2-1

x2-2x+1

ab2+3a3b-5ab

x3+1

1a+b+1a-b

x2-1x+1

Expand

The expand command is used mainly to rewrite polynomials with all brackets and whole number powers multiplied out and all like terms collected together. In the advanced section, you also have the option of expanding trigonometric functions, expanding modulo any integer and leaving certain parts of the expression untouched whilst expanding the rest.

Go to the Expand page

Factor

The factor command will try to rewrite an expression as a product of smaller expressions. It takes care of such things as taking out common factors, factoring by pairs, quadratic trinomials, differences of two squares, sums and differences of two cubes, and a whole lot more. The advanced section includes options for factoring trigonometric functions, factoring modulo any integer, factoring over the field of Gaussian integers (just the thing for those tricky sums of squares), and even extending the field over which factoring occurs with your own custom extensions.

Go to the Factor page

Simplify

Simplifying is perhaps the most difficult of all the commands to describe. The way simplification is performed in QuickMath involves looking at many different combinations of transformations of an expression and choosing the one which has the smallest number of parts. Amongst other things, the Simplify command will take care of canceling common factors from the top and bottom of a fraction and collecting like terms. The advanced options allow you to simplify trigonometric functions or to instruct QuickMath to try harder to find a simplified expression.

Go to the Simplify page

Cancel

The cancel command allows you to cancel out common factors in the denominator and numerator of any fraction appearing in an expression. This command works by canceling the greatest common divisor of the denominator and numerator.

Go to the Cancel page

Partial Fractions

The partial fractions command allows you to split a rational function into a sum or difference of fractions. A rational function is simply a quotient of two polynomials. Any rational function can be written as a sum of fractions, where the denominators of the fractions are powers of the factors of the denominator of the original expression. This command is especially useful if you need to integrate a rational function. By splitting it into partial fractions first, the integration can often be made much simpler.

Go to the Partial Fractions page

Join Fractions

The join fractions command essentially does the reverse of the partial fractions command. It will rewrite a number of fractions which are added or subtracted as a single fraction. The denominator of this single fraction will usually be the lowest common multiple of the denominators of all the fractions being added or subtracted. Any common factors in the numerator and denominator of the answer will automatically be cancelled out.

Go to the Join Fractions page

The notion of correspondence is encountered frequently in everyday life. Forexample, to each book in a library there corresponds the number of pages in thebook. As another example, to each human being there corresponds a birth date. Tocite a third example, if the temperature of the air is recorded throughout aday, then at each instant of time there is a corresponding temperature.

The examples of correspondences we have given involve two sets X and Y. Inour first example, X denotes the set of books in a library and Y the set ofpositive integers. For each book x in X there corresponds a positive integer y,namely the number of pages in the book. In the second example, if we let Xdenote the set of all human beings and Y the set of all possible dates, then toeach person x in X there corresponds a birth date y.

We sometimes represent correspondences by diagrams of the type shown inFigure 1.17, where the sets X and Y are represented by points within regions ina plane. The curved arrow indicates that the element y of Y corresponds to theelement x of X. We have pictured X and Y as different sets. However, X and Y mayhave elements in common. As a matter of fact, we often have X = Y.

Our examples indicate that to each x in X there corresponds one and only oney in Y; that is, y is unique for a given x. However, the same element of Y maycorrespond to different elements of X. For example, two different books may havethe same number of pages, two different people may have the same birthday, andso on.

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In much of our work X and Y will be sets of real numbers. To illustrate, let Xand Y both denote the set R of real numbers, and to each real number x let usassign its square x2. Thus to 3 we assign 9, to - 5 we assign 25, andso on. This gives us a correspondence from R to R. All the examples ofcorrespondences we have given are functions, as defined below.

Definition

A function f from a set X to a set Y is a correspondence that assigns to eachelement x of X a unique element y of Y. The element y is called the image of xunder f and is denoted by f(x). The set X is called the domain of the function.The range of the function consists of all images of elements of X.

Earlier, we introduced the notation f(x) for the element of Y whichcorresponds to x. This is usually read "f of x." We also call f(x) the value off at x. In terms of the pictorial representation given earlier, we may nowsketch a diagram as in Figure 1.18. The curved arrows indicate that the elementsf(x), f(w), f(z), and f(a) of Y correspond to the elements x, y, z and a of X.Let us repeat the important fact that to each x in X there is assigned preciselyone image f(x) in Y; however, different elements of X such as w and z in Figure1.18 may have the same image in Y.

Beginning students are sometimes confused by the symbols f and f(x). Rememberthat f is used to represent the function. It is neither in X nor in Y. However,f(x) is an element of Y, namely the element which f assigns to x. Two functionsf and g from X to Y are said to be equal, written

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for every x in X.

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Example 1 Let f be the function with domain R such that f(x) = x2for every x in R. Find f(-6) and f(a), where a is any real number. What is therange of f?

Solution Values of f(or images under f) may be found by substituting for x in theequation f(x) = x2 . Thus:

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If T denotes the range off, then by previous definition T consists of allnumbers of the form f(a) where a is in R. Hence T is the set of allsquares a2, where a is a real number. Since the square of any realnumber is nonnegative. T is contained in the set of all nonnegative realnumbers. Moreover, every nonnegative real number c is an image underf, sinceSolve equations, simplify expressions with Step-by-Step Math Problem Solver (5). Hence the range of f is the set of all nonnegative real numbers.

If a function is defined as in the preceding example, the symbol used for thevariable is immaterial; that is, expressions such as:

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and so on, all define the same function. This is true because if a is anynumber in the domain of f, then the same image a2 is obtained nomatter which expression is employed.

Example 2 Let X denote the set of nonnegative real numbers and let f bethe function from X to R defined bySolve equations, simplify expressions with Step-by-Step Math Problem Solver (7) for every x in X. Find f(4)and f(pi). If b and c are in X, find f(b + c) and f(b) + f(c).

Solution As in Example 1, finding images under f is simply a matter ofsubstituting the appropriate number for x in the expression for f(x). Thus:

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Many formulas which occur in mathematics and the sciences determinefunctions. As an illustration, the formula A = pi*r2 for the area Aof a circle of radius r assigns to each positive real number r a unique value ofA. This determines a function f, where f(r) = pi*r2, and we may writeA= f(r). The letter r, which represents an arbitrary number from the domain off,is often called an independent variable. The letter A, which represents a numberfrom the range off, is called a dependent variable, since its value depends onthe number assigned tor. When two variables r and A are related in this manner,it is customary to use the phrase A is a function of r. To cite another example,if an automobile travels at a uniform rate of 50 miles per hour, then thedistance d (miles) traveled in time t (hours) is given by d = 50t and hence thedistance d is a function of time t.

We have seen that different elements in the domain of a function may have thesame image. If images are always different, then, as in the next definition, thefunction is called one-to-one.

Solve equations, simplify expressions with Step-by-Step Math Problem Solver (2024)

FAQs

How do you simplify expressions step by step? ›

Step 1: Solve parentheses by adding/subtracting like terms inside and by multiplying the terms inside the brackets with the factor written outside. For example, 2x (x + y) can be simplified as 2x2 + 2xy. Step 2: Use the exponent rules to simplify terms containing exponents. Step 3: Add or subtract the like terms.

What is the website that solves any math problem? ›

Wolfram|Alpha has broad knowledge and deep computational power when it comes to math. Whether it be arithmetic, algebra, calculus, differential equations or anything in between, Wolfram|Alpha is up to the challenge.

How to solve math problems step by step? ›

Devise a plan
  1. Look for a pattern. Sometimes, the best way to solve a problem is to look for a pattern. ...
  2. Guess and check. ...
  3. Working backwards. ...
  4. Use a formula. ...
  5. Eliminate the possibilities. ...
  6. Use direct reasoning. ...
  7. Solve a simpler problem. ...
  8. Solve an equation.
Jun 3, 2022

How to solve a equation step by step? ›

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value. Combine like terms. Simplify the equation by using the opposite operation to both sides. Isolate the variable on one side of the equation.

How to solve simplify in mathematics? ›

How to simplify expressions. To simplify expressions first expand any brackets, next multiply or divide any terms and use the laws of indices if necessary, then collect like terms by adding or subtracting and finally rewrite the expression.

What is simplify expressions in math? ›

Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do.

What app gives you the answers to math problems? ›

Photomath is known worldwide for helping millions of learners to learn, practice, and understand math – one step at a time. Scan any math problem with the Photomath app to get step-by-step explanations with accurate solutions and a variety of teacher-approved methods.

Is there an AI that can solve math problems? ›

Google DeepMind's new AI systems can now solve complex math problems.

What is the best math website for free? ›

Top 10 Free Math Websites of All Time
  • Cliffs Notes.
  • Khan Academy.
  • Art of Problem Solving.
  • SumDog.
  • Greg Tang Math.
  • Illuminations.
  • Cool Math.
  • Hot Math.

What is the steps by step to solve the problem? ›

8-Step Problem Solving Process
  1. Step 1: Define the Problem. What is the problem? ...
  2. Step 2: Clarify the Problem. ...
  3. Step 3: Define the Goals. ...
  4. Step 4: Identify Root Cause of the Problem. ...
  5. Step 5: Develop Action Plan. ...
  6. Step 6: Execute Action Plan. ...
  7. Step 7: Evaluate the Results. ...
  8. Step 8: Continuously Improve.

What is the best free online calculator with steps? ›

Maple Calculator is a free powerful math solver and versatile math learning tool that gives you answers, 2-D and 3-D graphs, and even step-by-step solutions! Whether you are doing simple calculations or working on university-level math problems, Maple Calculator can do it all.

How can I solve math so fast? ›

6 Mental Math Strategies | Tips and Tricks for Students
  1. Rounding up to the nearest ten. ...
  2. Work from left to right. ...
  3. Use multiplication hacks. ...
  4. Bump the decimal over to easily find a percentage. ...
  5. Make guesstimates. ...
  6. Break down the problem.
Nov 9, 2020

What denotes equality between two expressions? ›

The equal sign or equal sign, formerly known as the equality sign, is the mathematical symbol =, indicating equality in some well-defined function. For example, in an equation, it is located between two expressions with the same value, or for which one analyses the conditions under which they have the same value.

What is a mathematical statement equating two quantities called? ›

An equation is a mathematical statement that two expressions are equal. The solution of an equation is the value that when substituted for the variable makes the equation a true statement.

What is the rule for simplifying expressions? ›

When we simplify an expression we operate in the following order:
  • Simplify the expressions inside parentheses, brackets, braces and fractions bars.
  • Evaluate all powers.
  • Do all multiplications and division from left to right.
  • Do all addition and subtractions from left to right.

In what order do you simplify expressions? ›

Expressions should always be simplified following the order of operations. For most simple expressions, that means simplifying the parentheses using the distributive property and then adding the like terms from left to right. Doing this out of order, leads to the incorrect answer.

What is the order in which steps must be done to simplify an expression? ›

The order is PEMDAS: Parentheses, Exponents, Multiplication, and Division (from left to right), Addition and Subtraction (from left to right). Is there a trick we can use to remember the order of operations? Yes. You can use the phrase “Please Excuse My Dear Aunt Sally” to remember PEMDAS.

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