MathJax Equation Builder

Here I'm testing a web app using what I learned in Dart flight school. I built the app and then uploaded the build folder to Google docs... embedded below in an iframe. Very easy using angular dart. Some LaTeX to try follows:

new line = \\  fraction = \frac{x}{y}  square root = \sqrt{x}

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Algebra Solving Worksheets

Solving is a very important skill to learn in any algebra course. The ability to solve a variety of different types of equations and inequalities takes time and practice to master.  Below are some worksheets that will help master these skills

[ Ch. 2 Solving Linear Equations ]
[ Ch. 2 Solving Linear Inequalities ]
[ Ch. 6 Solve Quadratic Equations by Factoring ]
[ Ch. 7 Solving Rational Equations ]
[ Ch. 8 Solving Radical Equations ]

If you are an instructor, please feel free to use these worksheets in your algebra course. More supplementary material can be found on our Elementary Algebra Textbook page.

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Algebra: Solving Quadratic Equations by Factoring

Previously we have learned how to solve linear equations, ax + b = 0, now we will outline a technique that can be used to solve factorable quadratic equations of the form ax^2 + bx + c = 0. Unlike linear equations, which usually have one solution, quadratic equations can have up to two solutions. We begin with the zero product property:

This property is the key to solving quadratic equations by factoring. Basically, if the quadratic equation is equal to zero you factor it, then set each factor equal to zero, and solve.

Tip: You can always check your answers by substituting them into the original equation to see if a true statement results. 

  

Solve.

Note: When one solution is obtained, as in the previous problem, we say that the solution is a double root. This technique requires the zero product property, so you first make sure that the quadratic equation is set equal to zero before factoring. Do this by adding all terms to one side of the equation. The check is usually optional.

A common mistake is to set each factor to 56 here - which leads to incorrect results. You must have zero on one side of the equation for this technique to work.

Solve.

Clearing fractions from our equations can be done by multiplying both sides of our equation by the LCM of the denominators. After doing this, the equation becomes a bit easier to solve.

Solve.

Finding equations given the solutions requires you to work the entire process in reverse. Given the solutions you can find factors that can be multiplied to obtain a quadratic equation.

Find a quadratic equation with the given solution set.

You can find much more in the free to read online open Elementary Algebra textbook available on the Flat World Knowledge website:

[ Free Algebra Textbook ]

UPDATE: Here are some corresponding exercises. Enjoy.

[ Exercises PDF ] [ Download Source DOCX ]

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Algebra: How to Solve Linear Systems

In our Algebra course, we are currently learning how to solve systems of two linear equations with two variables.  This topic is covered in Chapter 4 of our open Elementary Algebra textbook published by Flat World Knowledge. To find the simultaneous solution of such a system, or point of intersection, we can use one of three methods.

1. The Graphing Method:  Rewrite the two equations in point-slope form ( y = mx + b), graph them on the same set of coordinates and then determine the point of intersection. A video example follows:



2. The Substitution Method:  This completely algebraic method, which requires that we first isolate one of the variables. We then substitute the result into the other equation.  Once we solve for one of the variables, back substitute to find the value of the other variable.  Remember that the answer is an ordered pair (x, y).



3. The Elimination Method:  Usually this is the method of choice.  The idea is to multiply one or both of the equations by appropriate numbers so that one of the variables will eliminate if the equations are added together.  This is sometimes called the "addition method."  Always back substitute to find the value for the other variable and present the solution as an ordered pair.



Each method has its strengths and weaknesses, but whichever method you choose the answer will be the same.  Most of the time, a system will produce one solution, a single point of intersection.  However, sometimes the equations are actually equivalent and in that case there are infinitely many simultaneous solutions.  This describes a dependent system and the algebraic methods will lead to a true statement like 5 = 5 or 0 = 0. Click here for a video example:

It is also important to note that there is not always an answer to some linear systems.  Sometimes the lines are parallel and do not intersect. This describes an inconsistent system and the algebraic methods will lead to a false statement like 0 = 4.  Click here for a video example.











These take time and practice to master.  Try some yourself and you will soon see that solving systems of two linear equations is actually fun.  Also, be sure to look at the many applications found in section 4.4. Hope this helps.

UPDATE:  More Solving Linear Systems videos.  Visually searchable, click on a problem and view it worked out on youTube.

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Solving Rational Equations in Beginning Algebra

A thorough explanation for solving rational equations can be read for free in the Elementary Algebra textbook that is available on the Flat World Knowledge website:

[ Solving Rational Equations ] 

Also, you can click on the following rational equation to watch a video solution.

The basic steps for solving these equations are as follows:

Step 1: Factor all denominators and determine the LCD.

Step 2: Identify the restrictions. 

Step 3: Multiply both sides of the equation by the LCD. Distribute carefully and then simplify.

Step 4: Solve the resulting equation. 

Step 5: Check for extraneous solutions.

The complete solution is presented below:

Basically, we multiply both sides of the equation by the LCD as a means to clear the fractions.  This introduces the possibility for extraneous solutions and so we must check our answers.  Look for more Chapter 7 videos here.

[ Visually Searchable Algebra Problems ] 

Elementary Algebra (Ch. 9) Solving Quadratic Equations and Graphing Parabolas

Elementary Algebra is an open textbook that is available to read for free on the flatworld knowledge website. Chapter 9 gives an introduction to solving general quadratic equations and graphing quadratic functions. In addition, complex numbers and complex solutions are covered.

9.1 Extracting Square Roots
9.2 Completing the Square
9.3 Quadratic Formula
9.4 Guidelines for Solving Quadratic Equations and Applications
9.5 Graphing Parabolas
9.6 Introduction to Complex Numbers and Complex Solutions

You will find a complete review exercise section and a practice test.

9.7 Review Exercises and Sample Exam

Feel free to copy and paste the links found here and post them in your LMS. Links to all of the chapters can be found by clicking the Elementary Algebra course button at the top of this page.

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Elementary Algebra (Ch. 8) Radical Expressions and Equations

Elementary Algebra is an open textbook that is available to read for free on the flatworld knowledge website. Chapter 8 gives an introduction to radical expressions and equations.

8.1 Radicals
8.2 Simplifying Radical Expressions
8.3 Adding and Subtracting Radical Expressions
8.4 Multiplying and Dividing Radical Expressions
8.5 Rational Exponents
8.6 Solving Radical Equations

You will find a complete review exercise section and a practice test.

8.7 Review Exercises and Sample Exam

Feel free to copy and paste the links found here and post them in your LMS. Links to all of the chapters can be found by clicking the Elementary Algebra course button at the top of this page.

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Solve by Factoring in Elementary Algebra

Learning how to solve equations is one of our main goals in Algebra. At this point, we know how to solve linear equations,

And now we begin to learn how to solve quadratic equations,

One way to solve quadratic equations is to use factoring and the zero-product property. The zero-product property sounds fancy but it simply states that,

So if we can factor a quadratic equation in standard form, equal to zero, we can then set each factor equal to zero and solve the resulting linear equations. For example, solve the following quadratic equation:

Step 1: Express the quadratic equation in standard form, equal to zero.

Step 2: Factor the quadratic expression.

Step 3: Apply the zero-product property and set each variable factor equal to zero.

Step 4: Solve the resulting linear equations.

The solutions are -5, and -3/2. You can check that these are solutions by substituting them into the original equation and verify that they lead to a true statement.  Here is another example on video:

Hopefully you can see that the factoring step is very important in this process. Take care to factor your quadratic equations carefully because this is the step where most students make an error.

• Elementary Algebra Textbook - Chapter 6 Factoring and Solving by Factoring
• Elementary Algebra Textbook - Chapter 6 YouTube Playlist

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