# PDF Algebra

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### Solve problems from Pre Algebra to Calculus step-by-step

Integer Exponents — In this section we will start looking at exponents. We will give the basic properties of exponents and illustrate some of the common mistakes students make in working with exponents. Examples in this section we will be restricted to integer exponents. Rational exponents will be discussed in the next section. Rational Exponents — In this section we will define what we mean by a rational exponent and extend the properties from the previous section to rational exponents. We will also discuss how to evaluate numbers raised to a rational exponent.

Radicals — In this section we will define radical notation and relate radicals to rational exponents. We will also give the properties of radicals and some of the common mistakes students often make with radicals. We will also define simplified radical form and show how to rationalize the denominator. Polynomials — In this section we will introduce the basics of polynomials a topic that will appear throughout this course. We will define the degree of a polynomial and discuss how to add, subtract and multiply polynomials.

Factoring Polynomials — In this section we look at factoring polynomials a topic that will appear in pretty much every chapter in this course and so is vital that you understand it. We will discuss factoring out the greatest common factor, factoring by grouping, factoring quadratics and factoring polynomials with degree greater than 2. Rational Expressions — In this section we will define rational expressions. We will discuss how to reduce a rational expression lowest terms and how to add, subtract, multiply and divide rational expressions.

Complex Numbers — In this section we give a very quick primer on complex numbers including standard form, adding, subtracting, multiplying and dividing them. Solutions and Solution Sets — In this section we introduce some of the basic notation and ideas involved in solving equations and inequalities.

## Algebra I | Math | Khan Academy

We define solutions for equations and inequalities and solution sets. Linear Equations — In this section we give a process for solving linear equations, including equations with rational expressions, and we illustrate the process with several examples. In addition, we discuss a subtlety involved in solving equations that students often overlook. Applications of Linear Equations — In this section we discuss a process for solving applications in general although we will focus only on linear equations here.

Equations With More Than One Variable — In this section we will look at solving equations with more than one variable in them. These equations will have multiple variables in them and we will be asked to solve the equation for one of the variables. This is something that we will be asked to do on a fairly regular basis. Quadratic Equations, Part I — In this section we will start looking at solving quadratic equations. Specifically, we will look at factoring and the square root property in this section.

We will use completing the square to solve quadratic equations in this section and use that to derive the quadratic formula. The quadratic formula is a quick way that will allow us to quickly solve any quadratic equation. Quadratic Equations : A Summary — In this section we will summarize the topics from the last two sections. We will give a procedure for determining which method to use in solving quadratic equations and we will define the discriminant which will allow us to quickly determine what kind of solutions we will get from solving a quadratic equation.

Applications of Quadratic Equations — In this section we will revisit some of the applications we saw in the linear application section, only this time they will involve solving a quadratic equation. Equations Reducible to Quadratic Form — Not all equations are in what we generally consider quadratic equations. However, some equations, with a proper substitution can be turned into a quadratic equation. These types of equations are called quadratic in form. In this section we will solve this type of equation. Equations with Radicals — In this section we will discuss how to solve equations with square roots in them.

As we will see we will need to be very careful with the potential solutions we get as the process used in solving these equations can lead to values that are not, in fact, solutions to the equation. Linear Inequalities — In this section we will start solving inequalities. We will concentrate on solving linear inequalities in this section both single and double inequalities.

We will also introduce interval notation.

Polynomial Inequalities — In this section we will continue solving inequalities. However, in this section we move away from linear inequalities and move on to solving inequalities that involve polynomials of degree at least 2. Rational Inequalities — We continue solving inequalities in this section. Absolute Value Equations — In this section we will give a geometric as well as a mathematical definition of absolute value.

We will then proceed to solve equations that involve an absolute value. We will also work an example that involved two absolute values. Absolute Value Inequalities — In this final section of the Solving chapter we will solve inequalities that involve absolute value.

Graphing — In this section we will introduce the Cartesian or Rectangular coordinate system. We will illustrate these concepts with a couple of quick examples Lines — In this section we will discuss graphing lines. We will introduce the concept of slope and discuss how to find it from two points on the line.

### Determining Equations Based on Variables Through the Use of Formulas

To see what this looks like in practice, read on! How much change are they each left with? It turns out we can use a simple algebraic equation to figure this problem out quickly and easily. For this example, we are going to represent the change they are left with using the letter x any letter will do — x just happens to be a popular choice. The total change will, therefore, be represented by 2x because they need to divide it between themselves.

There are 26 cats in the pet shop. This is two more than three times the number of dogs. How many dogs are there in the pet shop? Alice and Bob have some candy. If Alice gives Bob a piece of candy, Bob has twice as much candy as her. If Bob gives Alice a piece of candy, they have the same number.

## Introducing Algebra

How many pieces of candy do they each have? When getting your child to attempt these questions, always ask them to show their reasoning, as this is the essence of algebra and indeed of all mathematics! In the last example, they may solve the problem through trial and error. Algebra becomes really powerful when it gives us systematic methods that can be applied to a flexible range of problems.

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What is Algebra? What is algebra? Where does the word algebra come from? So, why do children need to know algebra then? Want something a bit more taxing?