Your Essential Guide to Solving Quadratic Equations Pt I

Do quadratic equations seem a little confusing for you? Don’t worry!! We’ve got you covered with our two-part essential guide to solving quadratic equations 🙂

Introduction

Solving quadratic equations is an essential skill that you’ll need in Mathematics, including Maths in Years 9 and 10, Maths in Year 11 and particularly Maths in your HSC. In Year 12 Maths courses, not only will you be assessed on how well you can solve an equation algebraically, but you’ll also be required to apply your understanding of quadratic equations to topics such as trigonometry. Learning how to solve quadratic equations now will set you up quite nicely for success from Year 9 and beyond. 

The next two posts will provide an in-depth summary of quadratic equations. In Part I, we’ll start off by defining what a ‘quadratic equation’ is and will go through some common terminology. Then we’ll get into our first method: Solving quadratic equations through factorisation. Part II of our guide will cover two additional methods for solving quadratic equations: The quadratic formula and completing the square. 

Let’s dive straight in!!

What is a Quadratic Equation?

Overview

First let’s examine what a “quadratic equation” actually is:

• A quadratic equation is where the highest power or “index” of our x variable is 2.
• Examples of quadratic equations include x2+5x+4=0 and -3×2+9=0. What is important here is that there are no other x terms with an index greater than 2. 
• If we had an equation such as x3+x2-1=0, this would not be a quadratic equation because there is one x term with an index of 3. 
• Quadratic equations will generally have two answers, however, sometimes they can have one answer or no answers.

Now that we know what a quadratic equation is, let’s look at the ‘form’ of a quadratic equation:

• Quadratic equations can be expressed in the form ax2+bx+c=0. You can think of this as a “template” for quadratics, where the numbers for a,b and c are changed to produce different equations. 
• For example, 5×2+3x-1=0 is in the form above. In this equation, a=5,b=3 and c=-1.
• x2-3x+1=0 is another example. Here, a=1,b=-3 and c=1.

What about “monic” and “non-monic” quadratics?

• The difference all comes back to our “template” ax2+bx+c=0 and the value that the coefficient a takes on. 
• With monic quadratics, a always has a value of 1. Examples of monic quadratics include x2+3x+2=0 and x2-5x+4=0 (although we don’t write the “1” in front of the x term!!). 
• With non-monic quadratics, a can take on any other value apart from 1. Examples include 2×2-3x+9=0 and 0.5×2-4x-2=0.

So now that we have a good idea of (1) what a quadratic equation is, (2) the form in which they are expressed, and (3) the difference between monic and non-monic quadratics, we’ll now look at three methods of solving quadratic equations. The three methods we’ll cover include the following:

• Factorisation
• Quadratic Formula
• Completing the Square

In this post we’ll look at the first method: Factorisation. 

Factorisation

What is “Factorisation”?

The whole idea behind factorisation is to break down the quadratic equation into its essential factors. After this, we’ll then proceed to write the equation in brackets. From that point onwards, we can then solve for x. Here is a flowchart representing the process:

Equation   ->   Factorisation   ->   Solve

x2+3x+2=0 -> (x+1)(x+2)=0 -> x=-1;x=-2

Now, one important thing to note is that factorisation will only work where our equation is equal to zero. If you have terms on the other side of the equation, you will need to bring them over so the other side has nothing left. 

Factorising Monic Quadratics

Let’s look at the example above: x2+3x+2=0 (nothing that it follows the form ax2+bx+c=0). To “factorise” this equation, we need to think of two numbers that have a sum of the b term (in this case, 3) and a product of the c term (in this case, 2). To repeat:

_ + _ =3

_ _ =2

In this case, our two numbers are 2 and 1. That’s because:

2+1=3

21=2

So now that we have our two factors, we’ll now replace 3x with x+2x (which is equal to 3x) in the equation we’re solving. We get:

x2+x+2x+2=0

Now we can see that we have two “pairs” of terms: x2+x and 2x+2. The x can be factored out of x2+x to get x(x+1), and the 2 can be factored out of 2x+2 to get 2(x+1). So we get:

x(x+1)+2(x+1)=0

Now, what’s really important here is that both “pairs” of terms now have a common factor: (x+1). This is the stage we want to be at. Why: Because we can now factor (x+1) out of both “pairs” and put the outside terms x and 2 into another set of brackets to get a nicely packaged quadratic factorisation!!

(x+1)(x+2)=0

Now that we’ve done all the hard work, what’s left for us to do is to solve for x. We have two equations: x+1=0 and x+2=0. Let’s solve both:

x+1=0 x+2=0

x=-1 x=-2

And there you have it: x=-1 and x=-2!! 

Factorising Non-Monic Quadratics

What happens if we need to factorise a non-monic quadratic such as 3×2+5x+2=0? Well, the process is identical to the one we used for monic quadratics bar one exception.

Read carefully:

• We still need to find two numbers, both of which add to the b term (in this case, 5).
• However, the product changes. Rather than both numbers needing to multiply to give the c term (in this case 2), this changes so that the product is now the a term multiplied by the c term. In this case, the two numbers we find would need to multiply to give 32=6.

To repeat:

_ + _ =5

_ _ =6

In this case, our numbers are:

3+2=5

32=6

After this, we then do the same thing that we did for our monic quadratic above: That is, write 5x as 3x+2x:

3×2+3x+2x+2=0

Now we have the pairs 3×2+3x  and 2x+2 which can be factorised as 3x(x+1) and 2(x+1) respectively:

3x(x+1)+2(x+1)=0

Then factorise again:

(x+1)(3x+2)=0

Then solve:

x+1=0 3x+2=0

x=-1 x=-23

Done!!

What Happens When We Can’t Factorise? 

We might have some instances of where we cannot factorise a quadratic equation. For example, x2-3x-1=0 would need two numbers that add to give -3 and multiply to give -1. There aren’t any. 

In this case, we would check to see if the quadratic equation can be solved using a different method. We can either use the completing the square method or the quadratic formula. 

Conclusion

So there you have it: Part I of our essential guide to solving quadratic equations. To recap, we have looked at what a quadratic equation is, the ‘form’ of a quadratic equation, how to differentiate between monic and non-monic quadratics, and how to solve quadratic equations through factorisation. In our next post, we’ll explore  the quadratic formula and the “completing the square” method.