The Ultimate Guide: Quadratic Equations Made Easy

Do you start hating math when doing questions on quadratic equations? Well…

After reading our ultimate step-by-step guide you will understand:

• What quadratic equations are and how to approach them with ease, every time.
• The 3 methods that allow you to factorise ANY quadratic equation, with examples.
• Why factorising and solving quadratic equations is an essential skill in Year 11 and 12 mathematics (this isn’t just about factorising quadratic equations).
• BONUS: how to sketch a quadratic function (example and checklist included).

Concept Check

Before we begin, you should already know:

• What factorising is.
• How to factorise simple linear equations; i.e., ab + bc = a (b + c).
• How to expand and factorise using perfect squares and difference of two squares: i.e.,

(a + b)² = a²+ 2ab + b²

(a-b)² = a² – 2ab + b²

(a + b)(a – b)= a² – b²

Warning: if you never want to struggle with quadratics again, read the blog from start to finish.

Are you ready? Let’s go.

Introduction

A quadratic equation is anything that is in the form,

y = ax² + bx + c

Where:

• a is called the leading coefficient
• b is called the linear coefficient
• c is called the constant term.

Notice that the highest power of x in a quadratic equation is 2 (i.e., the power of x²).

Quick Example: In the quadratic equation,  y = 3x²+ 7x + 4, a = 3 b = 7 c = 4

Keep in mind that a, b and c can be any number, with one exception: a ≠ 0. 

If a = 0, the equation would simplify to:

y = bx + c

This is now a linear equation (it is now in the form y = mx + b) and it is no longer quadratic.

Quick Example: Find a, b and c for the following quadratic equation y = x² – 5x + 6, We can rewrite the quadratic equation as follows: y = x² – 5x + 6 y = (1) x² + (-5)x + (6). a = 1 b = -5 c = 6.

In a quadratic equation, 

• When a=1, the equation is known as a monic quadratic; monic comes from the Greek word mono meaning one; which in this case is the value of a.
• When a1, the equation is known as a non-monic quadratic.

y = x² – 5x + 6 (monic quadratic)

 y = 3x² + 7x + 4 (non-monic quadratic)

In the above non-monic quadratic, a = 3.

We often like to solve quadratic equations, meaning we like to find what values of x (and y) satisfy the equation.

Quick Example: Factorise and solve the quadratic equation x² – 2x = 0 Factorising: making the equation cleaner and easier to analyse.. x² – 2x = 0 x (x – 2) = 0 Solving: finding values of x that satisfy the equation. If x = 0, 0(0 – 2) = (0)(-2) = 0 If x = 2, 2(2 – 2) = (2)(0) = 0 Therefore the values of x that satisfy this equation are x = 0 and x = 2.

The x values that satisfy the equation are called the roots of the quadratic equation.

You can see that factorising the quadratic equation first made it easier for us to solve it. This is why we put a lot of focus on factorising methods, especially for more challenging examples.

Solving it allows us to sketch the quadratic equation (quadratic function) on a cartesian plane; the shape of it will be a parabola.

But first, let’s take a look at…

The 3 Methods That Can Factorise ANY Quadratic Equation

1. The Shortcut Method

Different people like to call this method different names: cross method, pairing method, fraction method, etc. These methods vary slightly but are essentially doing the same thing. So forget about the name and let’s learn what the steps are. Here we go…

Suppose we are looking at a quadratic in the general form:

y = ax² + bx + c.

Here the goal is to represent the quadratic equation in the factored form:

y = (x – α)(x – β)

Where α and β will end up being constants.

The shortcut method of factorising involves finding 2 numbers that multiply to get c and add to get β.

Let’s do an example…  

Monic Quadratics

Quick Example: Factorise the quadratic y = x² + 5x + 6. y = x² + 5x + 6 b = 5 c = 6. To factorise the quadratic, we need to find 2 numbers that multiply to get 6 and add to 5. Let’s break this into steps. Step 1: Look at the factors of 6. Factors of 6: ± 1, ± 2, ± 3, ± 6. (Note: there is no rule that states factors cannot be negative, so we include them as well.) Step 2: Look at which two factors add to 5. It’s easy to see that 2 and 3 add to give you 5. Step 3: Using the 2 numbers we just found, split the 5x into 3x + 2x, then factorise: y = x² + 5x + 6 y = x² + 3x + 2x + 6 y = x (x + 3) + 2 (x + 3) y = (x + 2)(x + 3) This is now the factorised form of the quadratic.

Using this factorising method, we can then solve quadratic equations.

Quick Example:  a. Factorise the quadratic y = x² – 5x + 6. b. Hence solve  x² – 5x + 6 = 0 a. Factorising: We can rewrite the quadratic equation as follows: y = x² – 5x + 6 b = -5 c = 6. Step 1: Look at the factors of 6. Factors of 6: ± 1, ± 2, ± 3, ± 6. Step 2: Look at which two factors add to -5. It’s easy to see that -2 and -3 add to give you -5. Step 3: Using the 2 numbers we just found, split the -5x into -3x-2x, then factorise: y = x² – 5x + 6 y = x² – 3x – 2x + 6 y = x(x – 3) – 2(x – 3) y = (x – 2)(x – 3) This is now the factorised form of the quadratic. b. Solving:  x² – 5x + 6 = 0 (x – 2)(x – 3) = 0 If x = 2, (2 – 2)(2 – 3) = (0)(-1)= 0 If x = 3, (3 – 2)(3 – 3)= (1)(0) = 0 Therefore the values of x that satisfy this equation are x = 2 and x = 3.

Non-Monic Quadratics

Notice that we have done 2 examples of factorising and solving a monic quadratic. For non-monic quadratics, the two numbers that you have to multiply and add to are slightly different. Here’s how you factorise a non-monic quadratic.

Quick Example: Factorise y = 3x² + 7x + 4 Preparation Step: Identify the value of b and a x c: b = 7 a x c = 3 x 4 = 12 Now we have to find two numbers that add to 7 and multiply to get 12. Step 1: First look at the factors of 12.Factors of 12: ±1, ±2, ±3, ±4, ±6, ±12. Step 2: Look at which two factors add to 7. It’s easy to see that 3 and 4 add to give you 7. Step 3: Now let’s go back to our quadratic equation, break the 7x into 3x+4x and factorise: y = 3x² + 7x + 4 y = 3x² + 3x + 4x + 4 y = 3x (x + 1) + 4 (x + 1) y = (x+1)(3x+4) This is the factorised form of y = 3x² + 7x + 4.

Now you know how to factorise monic and non-monic quadratics using the shortcut method.

The second method of factorising quadratics is….

2. Completing the Square

The method of completing the square is one that most students forget very quickly after learning it because it can be an overwhelming method at first. But don’t worry, we’ll make it easy peasy. We mainly use this method for quadratics that CANNOT be factorised into the form y = (x – α)(x – β). However, this method will still work on any quadratic.

Completing the square is all about factorising used perfect squares; i.e., a² ± 2ab + b² = (a ± b)².

The goal here is to turn a quadratic in general form into vertex form:

y = (x + h)² + k

Don’t worry why it’s called ‘vertex form’ for now, just learn how to complete the square..

Ok, let’s make this repeatable as possible so that you can perform the exact same steps every time you complete the square.

Let’s give it a shot…

Quick Example: Factorise y = x² + 3x + 7 by completing the square. y = x² + 6x + 7 Step 1: Let’s add and subtract When completing the square, ALWAYS write (3)2 rather than 9 . You’ll see why in a second. y = x² + 6x + (3)² – (3)² + 7 Step 2: Factorise the first 3 terms and simplify the last 2 terms; i.e., x² + 6x + (3)² = (x + 3)², and – (3)² + 7 = -2 So now we get: y = x² + 6x + (3)² – (3)² + 7 y = (x + 3)² – 2 This is the factorised form of y = x² + 3x + 7. You can see that we wrote (3)² rather than 9 because it is visually much easier to convert it to a perfect square. If we want to convert x² + 6x + (3)² into a perfect square, just identify the first and last squared terms (x and the +3) and now form your perfect square (x + 3)².

Let’s try an example which is not as easy as the one above. using the EXACT same steps.

Quick Example: a. Factorise the quadratic y = x² – 5x + 4 by completing the square. b. Hence solve  x² – 5x + 4 = 0. a) Factorising: y = x² – 5x + 4Step 1: Add and subtract Step 2: Factorise the first 3 terms and simplify the last 2 terms. Solving: Using the ‘completed square’ form that we just found, solving becomes easy. x² – 5x + 4 = 0 Therefore the values of x that satisfy this equation are

We don’t often complete the square of non-monic quadratics. But if you want to do so, you first convert it into a monic quadratic and complete the square from there, like in the following example.

Quick Example: Factorise y = 2x² + 3x + 2 by completing the square. Factorise the a = 2 to get a monic quadratic: Inside the brackets we have a monic quadratic. Now we can complete the square of the inside. The steps are exactly the same as in the two previous examples. Now let’s expand just to make the solution a little cleaner.

Now you know how to factorise using completing the square.

The final method of factoring quadratic equations is…

3. Factorising Using the Quadratic Formula

Suppose that we want to solve the equation:

0 = ax² + bx + c.

When solving any quadratic equation, the goal is to find x values that satisfy the equation. How do we turn this into an equation that has x on one side (i.e., x = something)?

Let’s factorise this equation by doing what we’ve just learnt, completing the square! After that we can solve it.

Deriving the Quadratic Formula: Let’s do some preparation by dividing every term by a: 0 = ax² + bx + c Now let’s try completing the square. We need to add and subtract Now let’s try solving by making x the subject. Now that we have x on one side, let’s do some cleaning by simplifying the equation. We start by getting a common denominator inside the square root. Now let’s put it all under one fraction. Done!

After completing the square and solving 0=ax²+bx +c, we get:

This equation is called the quadratic formula. It allows us to immediately solve almost any quadratic equation.

Let’s do an example…

Quick Example: Using the quadratic formula, solve  x2 – 5x + 6 = 0. a = 1 b = -5 c = 6 In the + case: In the – case: Therefore the values of x that satisfy this equation are x=2 and x=3. Let’s show this: (2)2 – 5(2) +6 = 0 (3)2 – (5(3) + 6) = 0 We did this same example using the shortcut method above and we got the same answer! Note that x = 2,3 are called the roots of the quadratic equation or the roots of the quadratic formula.

Suppose that we want to use the quadratic formula to factorise an equation.

Once we get our 2 solutions (x+ and x-), we want to express the equation in the form:

y=(x-x₊)(x-x_–)

Let’s try this using our previous example… 

Quick Example: Using the quadratic formula, solve  x2 – 5x + 6 = 0. a = 1 b = -5 c = 6 Now if we want to rewrite the equation in its factorised form, we simply write: y = (x – x+)(x-x–)y = (x – 3)(x – 2)

Let’s try one more…

Quick Example: Using the quadratic formula, solve  x2 – x – 2 = 0. a = 1 b = -1 c = -2 In the + case: In the – case: Now if we want to rewrite the equation in its factorised form, we simply write: y = (x – x+)(x-x–)

Make sure that if you get a negative result, remember to include the minus sign.

Quick Example: Using the quadratic formula, factorise  x² + 5x + 6 = 0 a = 1 b = 5 c = 6 In the + case: In the – case: Now if we want to rewrite the equation in its factorised form, we simply write: y = (x – x+)(x-x–)y = (x – (-2)) (x -(-3))y = (x + 2)(x + 3)

It’s that simple! Make sure you follow the same procedure each time, it’ll make life much easier when factorising and solving these types of equations.

In order for the quadratic formula,

to work in the world of real numbers, the inside of the square root cannot be negative, i.e., b² – 4ac ≥ 0. This is because we cannot do square roots of negative numbers; we would need complex numbers from extension 2 math to work with negatives inside square roots which is beyond the advanced and extension 1 syllabus.

Quadratics where b² – 4ac ≥ 0 cannot be factorised using the quadratic formula. Instead, completing the square is a favourable method.

Here’s a quick example…

Quick Example: Using the quadratic formula, factorise  x² + 5x + 8 = 0 a = 1 b = 5 c = 8 5² – 4(1)(8) = -7 < 0 Since b2-4ac<0, this quadratic cannot be factorised or solved using the formula. However you can still factorise it using completing the square to get the final result:

Now you know how to factorise and solve using the quadratic formula. If you’re not sure if you get a question correct, you can use an online quadratic formula calculator or a quadratic equation solver to check your answer.

Final Thoughts

That’s everything you need to know about factorising and solving quadratic equations! Here’s a quick summary below.

A quadratic equation is anything in the form y=ax2+bx+c. We like to factorise quadratic equations so that we can easily solve quadratics and sketch them on a cartesian plane with ease.

There are only 3 methods of factorising quadratic equations:

1. Shortcut Method.
2. Completing the Square.
3. Using Quadratic Formula.

You can factorise any quadratic equation using any of the 3 methods; except when b2-4ac0, in which case you can only factorise these quadratics using completing the square. When you factorise the equation, solving it becomes easy and the answer is only 2-3 short lines of writing away.

BONUS: Sketching Quadratic Functions.

It is assumed that you know what a parabola looks like and what the graph y=x2 looks like.

There are many components needed when sketching quadratic functions. Let’s keep things simple by learning the basics of sketching these functions by doing a quick example. We’ll use a quadratic equation that we’ve already seen before.

Quick Example: Sketch the quadratic y = x² – 5x + 6. We first need to factorise using the shortcut method, which we have already done previously. y = (x-2)(x-3) Now, just like when you sketch linear functions, you need first to find the x and y intercepts Finding the y-intercept: When x=0, y = 0² – 5(0) + 6 = 6y = 6 Finding the x-intercept: (this is where the solving part we’ve been doing comes in). When y=0, (x-2)(x-3) = 0(x-2)x = 2, 3. Now we can sketch it by first putting in the x and y intercepts and then connecting the dots:

Here’s a checklist when sketching quadratic functions. Make sure you:

1. Label your x and y axes,
2. Mark your x and y intercepts on your axes,
3. Draw the parabola in the correct location,
4. Make your parabola as ‘to-scale’ as possible.
5. Draw the parabola with the correct concavity (concave up/down).

Now you understand the basics of sketching quadratic functions.

It’s difficult to cover everything about quadratic functions in a single blog; further sketching of quadratic functions, quadratic inequalities, etc. This is why in our weekly classes we take things slow and break the content down into simple steps spread throughout the weeks with plenty of practice questions and done-for-you examples. If this blog was helpful, then you are invited to join a class for a week at no cost and no risk!

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Written by one of our superstar Maths tutors, Bartosz Mrowka. Bartosz received an ATAR of 97.8 and has completed a Bachelor of Advanced Science with a specialisation in Advanced Physics.