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How to Factorise Quadratic Expressions and Solve Quadratic Equations

Three lines of work showing how to solve the quadratic equation three x squared plus twenty five x minus eighteen equals zeroMany students struggle with analysing quadratic equations and graphing parabolas.  This page is devoted to explaining how to analyse quadratic equations.  Another page is devoted to the graphing of parabolas and explaining their properties.

You will find here a comprehensive series of videos and resource materials that will thoroughly explain how to analyse and factorise quadratic expressions and how to solve quadratic equations.

Simply put, a quadratic expression is one that can be resolved (simplified) into a maximum of three parts: a term consisting of a constant (number) multiplied by x², a term consisting of a constant (number) multiplied by x, and a term consisting of a number only.  Such an arrangement is called a simple polynomial.  Of the three terms, only the first one (containing the x²) is necessary for the expression to be called a quadratic expression.  This means that quadratic expressions may have one, two or three terms depending on whether they contain an “x” term and/or a constant.  Examples of quadratic expressions are: x² – 3x +5, 5x², -x² + 7, and 5x² -4x/3.  Quadratic expressions that are written in a messy way may be simplified, and one of your earlier challenges will be to learn how to do that.

Note that, although other numbers are permissible, on this page we will require all the constants to be real numbers (and most will be integers … whole numbers).

Quadratic equations are obtained by letting a quadratic expression equal another quadratic expression, or a linear expression (e.f. 2x – 5), or simply a constant.  Equations have an = sign, expressions do not.  We solve equations, because any quadratic equation may have up to two values for x that make the equation true.  All other values will fail to balance the equation.

I have said enough.  It will all make sense as we proceed …

FREE Excel Workbook to Help You Factorise Quadratic Expressions

Before you learn how to factorise quadratic expressions, you will need a supply of them (and their answers).  This is because constant practice can make you an accomplished 'factoriser,' and factorising is an essential skill in algebra and graphing.  Avoiding this practice is a bit like saying that you want to become a good sailor, surf skier, sail boarder or surfer, but you will not invest time in learning to swim.

During the last decade or so I have developed an Excel Workbook as a resource for my students. It produces a huge variety of expressions for you to practise on.  The quadratic expressions and equations are randomly generated, literally at the touch of a button (the F9 key). Also, the answers are provided on the same sheet! You can even change the level of difficulty at will.

Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Parents and students are free to use and distribute it freely. The only condition is that you understand that I retain copyright (ownership) of the workbook and I give you permission to distribute freely, but not to sell it in any way.

The workbook contains no macros and is virus free.  You may obtain it here.

If you find this workbook useful (or if you have suggestions for improvements), please leave a comment for me. Thank you.

How to Factorise Monic Quadratic Expressions Using the Cross Method

Factorising a quadratic expression is the complemetary (or reverse) process to expanding a binomial product. In fact, We use our understanding of expanding a binomial product to develop our strategies/methods to factorise quadratic expressions.  Mathematicians have devised a number of methods (or algorithms) for achieving this goal.

In this video (23:59) I explain the 'conventional' technique which is often called the Cross Method.

I will show you a clear way of presenting your work, and thinking through the process for finding the factors.

If you wish to practise these skills, you may use my FREE Excel workbook to produce a huge variety of expressions for you to practise on. In each case, the answers are provided on the same sheet! Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Please understand that I retain copyright and give you permission to distribute freely, but not to sell it in any way.

The above video is long (23:59).  If you do not wish to learn why the cross method works, or of you have already been taught this method and simply want to refresh your memory, this video is for you.  In it, I provide a short example (3:46) of factorising a quadratic expression using the cross method.

How to Factorise Monic Quadratic Expressions Using the Product-Sum (PS) Method

The Product-Sum Method underlies most methods for factorising quadratic expressions. The variations are mostly due to how the work is set out on the page.

Many students have difficulty with the concept, mainly because it is difficult to think in terms of positive and negative numbers. Here (15:18), I show a simple method to break the task into two simple steps ... first find the numbers and THEN decide whether they are positive or negative!  By explaining the method first, and then rapidly solving four examples, my hope is that you will be able to understand clearly how to use this method/system.

If you wish to practise these skills, you may use my FREE Excel workbook to produce a huge variety of expressions for you to practise on. In each case, the answers are provided on the same sheet! Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Please understand that I retain copyright and give you permission to distribute freely, but not to sell it in any way.

Mastering the Product-Sum Calculation for Factorising Quadratics

This is one of the most important videos in this series about factorising quadratic expressions. Regardless of the method that you use, at some stage you will need to find two numbers that multiply to give one result and add to give another result. We call this the product-sum. Many students find this process difficult, especially when the numbers required can be positive OR negative and particularly if they are weak with their number skills!

Why should you watch this video? Because, in it, I explain a method that you probably will not have seen! It allows you to find these two numbers in two simple steps instead on one, more complicated, step. I will say no more ... if you are rushed for time, start about halfway through the (15:09) video to see my demonstration (but you might find the entire video worth watching).

As I mention in the video, I have created an Excel workbook so that you can practise this skill and become very good at it! It allows you to practise ... first with positive numbers only and then with a mixture of positives and negatives. In each case, you can adjust the degree of difficulty and, by pressing the F9 key, create a randomly generated set of ten questions each time.  The answers are provided on the same sheet.

You can start by working on the sheet and, as you get better and better, solve them mentally. Fold the sheet in half so that a friend can see the answers and check you as you work down the page.

Teachers, feel free to provide a copy of this to each of your students and to use the sheets in class. I have used them very successfully in classes ... pairing students up so they hold the sheet between them and quickly test each other (or getting them to work on the sheets alone for more difficult sets). It helps students develop this particular skill very quickly. Please understand that I retain copyright and give you permission to distribute the file freely, but not to sell it in any way.

Factorising Non-Monic Quadratics Using the Cross Method

Monic quadratic expressions have an x² term with a coefficient of one. If you encounter a quadratic expression where the coefficient of x² is any number other than one, it is non-monic! These quadratics are more difficult to factorise.

Again, I start by demonstrating the conventional technique which is often called the Cross Method. I also explain why it works and mention its strengths and weaknesses.

The following videos will explain how to factorise the same quadratic expressions using the PS Method.

If you wish to practise these skills, you may use my FREE Excel workbook to produce a huge variety of expressions for you to practise on. In each case, the answers are provided on the same sheet! Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Please understand that I retain copyright and give you permission to distribute freely, but not to sell it in any way.

Factorising Non-Monic Quadratics Using Decomposition

The decomposition method for factorising quadratic expressions is given this name for a reason. We begin by using the product-sum technique to separate (or decompose) the middle term into two other terms ... hence the name. E.g. we would separate 7x² +4x - 3 into 7x² -3x +7x -3 (the term +4x having been decomposed into -3x +7x).

Once the quadratic has been separated into four terms, we factorise in pairs, looking for common factors. I demonstrate the technique thoroughly and explain why some people prefer this method above others (i.e. I explain some of the benefits to you). One of the major benefits is that, about half-way through the procedure you have a very good idea whether your answer is right or wrong, without having to expand your solution to test it! The major drawback is that this technique involves more writing than almost any other technique.

I encourage you to try this technique out. Factorise a few quadratic expressions this way to 'get a feel' for the process. Even if you decide this isn't for you, you will have gained from the experience!

If you wish to practise these skills, you may use my FREE Excel workbook to produce a huge variety of expressions for you to practise on. In each case, the answers are provided on the same sheet! Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Please understand that I retain copyright and give you permission to distribute freely, but not to sell it in any way.

Factorising Non-Monic Quadratics Using the Berry Method

I have used variants of this method for decades without knowing its name. If you know why it is called the Berry Method, please let me know. None of the sites that I have looked at on the Internet (including Wikipedia) seem interested in that detail.

In this video I demonstrate this clever method thoroughly using a number of examples.

Its advantages are that it requires only two lines of work (the quadratic is already factorised on the first line and the second line is basically 'tidying up') and there is no attempt to divide the coefficient of x² into factors during the more difficult first step.

I encourage you to try this technique out. Factorise a few quadratic expressions this way to 'get a feel' for the process. Even if you decide this isn't for you, you will have gained from the experience!

If you wish to practise these skills, you may use my FREE Excel workbook to produce a huge variety of expressions for you to practise on. In each case, the answers are provided on the same sheet! Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Please understand that I retain copyright and give you permission to distribute freely, but not to sell it in any way.

Factorising Non-Monic Quadratics Using the Diamond Method

Here I explain the Diamond Method for factorising quadratic expressions. It is, in fact, a variant of the Berry Method (see my last video/post). This time, instead of waiting to divide the factors after they have been constructed, all the division/simplifying is completed on the right hand side of the page where all the 'working' is done!

This means that, apart from the work on the right hand side of the page, the entire factorisation is completed on the one line!

I encourage you to try this technique out. Factorise a few quadratic expressions this way to 'get a feel' for the process. Even if you decide this isn't for you, you will have gained from the experience!

If you wish to practise these skills, you may use my FREE Excel workbook to produce a huge variety of expressions for you to practise on. In each case, the answers are provided on the same sheet! Teachers, feel free to provide a copy of this to each of your students. It will help them develop their skills quickly. You might even consider using it in class. Please understand that I retain copyright and give you permission to distribute freely, but not to sell it in any way.

Completing the Square With Simple (Monic) Quadratic Equations

The second step in analysing a quadratic equation is to find its roots (or zeros). These, of course, are the x-intercepts of the parabola.  These values can be found in three ways ... by factorising, by completing the square, and by using the Quadratic Formula.  All three methods do essentially the same job and find the same values.

I have posted about eight videos explaining how to factorise quadratic expressions. On another page, I will explain how to use this skill to find the roots/zeros of a quadratic equation and draw its graph.  You will be encouraged to know that, if you can factorise, you will have done the 'hard work.'

In this video (and the two that follow) I explain how to complete the square. I start by explaining what a perfect square looks like and how to use its pattern to solve a couple of equations.

The 'Completing the Square' videos are presented as follows:

  1. Monic quadratics with an even coefficient of x (this video/post)
  2. Monic quadratics with odd coefficient of x (the next videopost below this one)
  3. Non-monic quadratics (i.e. I will show you how to complete the square for any quadratics expression) (the next drop-down menu item below this one)

In this video/post I explain how to complete the square of a monic quadratic equation with an odd coefficient of x. These are a little bit more difficult than the ones with an even coefficient of x (which I examined in the last video/post).  This is because they require that we use fractions.  Many students today try to evaluate the parts as decimals, but that involves unnecessary calculation. I show here the benefits of using fractions, and how to solve such quadratic equations with relative ease.

This is a very important skill to develop, especially if you plan to study more advanced mathematics. Therefore, I strongly encourage you not to neglect it (even though it is under-emphasised in a lot of text books)! Practise this skill until you find it easy ... that is, until you can solve a batch of them at the rate of about one every 30 seconds. Then you will/can feel confident.

Completing the Square With Non-Monic Quadratic Equations (General Solution)

After having posted two videos showing how to complete the square for simpler quadratics expressions, I am now going to show you how to use this method with any quadratic expression. As you will have seen in the previous video, this solution is best completed using fractions rather than decimals.

If you are a high school student, it is more likely that you will be asked to simplify/solve a monic quadratic by completing the square. However, I encourage you not to neglect the general solution (shown here), especially if you plan to study more advanced mathematics. Set time aside to practise simplifying/solving non-monic quadratics as well. Even with the higher degree of difficulty that they present, you should be able to simplify/solve most of them in well under 60 seconds each. Then you will/can feel confident.

How to Learn the Quadratic Formula

Many students have difficulty with learning the quadratic formula.  It is the first 'complicated' formula that they learn in High School. I have found that students learn this formula best when they practise deriving it and using it.

If you wish to learn (and practise) how to derive the quadratic equation, download my PDF summary of the process and watch the video below.

If you wish to practise using the quadratic formula to solve lots of quadratic equations, I have created an Excel workbook that will allow you to generate randomised worksheets (and much more). You may download it here. Teachers will also find this workbook useful for creating practice and revision material for the classroom.

My personal preference is for students to learn by deriving and using the formula because, in this way, they know what they are learning (and why). However, if you simply want a method that 'drills the formula into your mind,' you might try one or more of the following:

  • Using rhythm (rap): The only example I found with reasonable sound quality was this one.
  • Using a story line with symbols: This one was particularly clever.

If you want to learn a bit about memory techniques, you might watch this TED Talk by Joshua Foer!

I encourage you to go exploring on YouTube if you wish to embark on a new adventure of developing your memory skills! Please note that I do not necessarily recommend any of these techniques. I suggest that you explore and find what works for you.

I enjoyed your presentation and no it wasn’t too long. Each subtraction algorithm has its merit as you demostrated, but after learning the “one up and one down” method, I’m employing it because of its speed and ease of usage. Even my wife, who hates mathematics with a passion, thinks it’s too easy. I look forward to your future presentations on both multiplication and number theory. I read an introduction text book some twenty five years ago on number theory by Oystein Ore who taught at Yale for better than twenty years. So in closing, please produce these lectures and the longer the better. Thanks.
Dennis Bell (on a CCM YouTube video about How to Subtract (Large) Numbers Easily)

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