Factorising Quadratics

Introduction

students, in this lesson you will learn how to factorise quadratics and why this skill matters in the study of functions 📘. A quadratic is a function of the form $f(x)=ax^2+bx+c$, where $a\neq 0$. Factorising means rewriting the quadratic as a product of simpler expressions, often of the form $a(x-p)(x-q)$. This is a powerful tool because it helps you find roots, sketch graphs, and solve equations more efficiently.

By the end of this lesson, you should be able to:

explain key terms such as quadratic, factor, root, zero, and intercept,
factorise quadratic expressions using appropriate methods,
connect factorising to graphing and solving equations,
see how quadratics fit into the wider Functions topic in IB Mathematics Analysis and Approaches SL.

A strong understanding of factorising quadratics is useful across mathematics and in real life. For example, if a ball follows a curved path, its height may be modelled by a quadratic function, and factorising can help find when it hits the ground 🏀.

What a Quadratic Looks Like

A quadratic function has the general form $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants and $a\neq 0$. The graph of a quadratic is a parabola. If $a>0$, the parabola opens upward; if $a<0$, it opens downward.

The values where the graph crosses the $x$-axis are called the roots or zeros of the function. These are the values of $x$ that make $f(x)=0$. If a quadratic can be written as $f(x)=a(x-p)(x-q)$, then the roots are $x=p$ and $x=q$.

This is why factorising is so important. It turns a polynomial expression into a product, making it easier to see the solutions. In functions language, factorising helps move between algebraic form and graphical meaning.

For example, consider $f(x)=x^2-5x+6$. If we factorise it as $f(x)=(x-2)(x-3)$, then we can immediately see that the roots are $x=2$ and $x=3$. This means the graph crosses the $x$-axis at $(2,0)$ and $(3,0)$.

Why Factorising Helps in Functions

In IB Mathematics Analysis and Approaches SL, functions are not just about calculating values. They are also about understanding structure, meaning, and behaviour. Factorising quadratics helps with all of these.

First, it helps solve equations such as $x^2-5x+6=0$. Instead of trying to guess the solutions, you can rewrite the expression as $(x-2)(x-3)=0$. Then use the zero-product property: if $AB=0$, then $A=0$ or $B=0$. So $x-2=0$ or $x-3=0$, giving $x=2$ or $x=3$.

Second, it helps with graph interpretation. The factorised form gives the $x$-intercepts directly. This is useful when sketching without a calculator, which is an important exam skill.

Third, factorising can help identify transformations and relationships between functions. For example, if $f(x)=x^2-4x+4$, then factorising gives $f(x)=(x-2)^2$. This shows the graph has a repeated root at $x=2$, so it touches the $x$-axis at one point instead of crossing it.

Methods for Factorising Quadratics

There are several common methods for factorising quadratics. The best method depends on the form of the expression.

1. Common factor first

Always check whether all terms share a common factor. For example,

$$f(x)=3x^2+6x=3x(x+2).$$

Here, $3x$ is the common factor. This step is important because it simplifies the expression before any further factorising.

2. Factorising monic quadratics

A monic quadratic has leading coefficient $1$, so it looks like $x^2+bx+c$. To factorise, find two numbers that multiply to $c$ and add to $b$.

Example:

$$x^2+7x+12$$

We need two numbers that multiply to $12$ and add to $7$. The numbers are $3$ and $4$, so

$$x^2+7x+12=(x+3)(x+4).$$

Check by expanding:

$$(x+3)(x+4)=x^2+4x+3x+12=x^2+7x+12.$$

3. Factorising non-monic quadratics

A non-monic quadratic has leading coefficient not equal to $1$, such as $2x^2+7x+3$. One useful method is the split-middle-term method.

Example:

$$2x^2+7x+3$$

We look for two numbers that multiply to $2\cdot 3=6$ and add to $7$. The numbers are $6$ and $1$.

Rewrite the middle term:

$$2x^2+6x+x+3$$

Now factor by grouping:

$$2x(x+3)+1(x+3)=(2x+1)(x+3).$$

Check by expanding:

$$(2x+1)(x+3)=2x^2+6x+x+3=2x^2+7x+3.$$

4. Difference of two squares after rearranging

Sometimes a quadratic can be written in a form that uses the identity $a^2-b^2=(a-b)(a+b)$. For example,

$$x^2-16=(x-4)(x+4).$$

This is not a full quadratic trinomial, but it is closely related to factorising quadratics and is often included in the same skill set.

Connecting Factorising to Roots and the Graph

One of the best reasons to factorise a quadratic is that it reveals the roots immediately. If

$$f(x)=a(x-p)(x-q),$$

then setting $f(x)=0$ gives

$$a(x-p)(x-q)=0.$$

Since $a\neq 0$, the roots are $x=p$ and $x=q$.

This has a clear graph meaning. The graph crosses the $x$-axis at the roots. If the quadratic has two different roots, the parabola usually crosses the $x$-axis twice. If it has one repeated root, the graph touches the $x$-axis and turns around. If it has no real roots, the graph does not meet the $x$-axis at all.

Example:

$$f(x)=(x-1)^2$$

Expanding gives

$$f(x)=x^2-2x+1.$$

This quadratic has a repeated root at $x=1$. The graph touches the axis at $(1,0)$.

Compare this with

$$f(x)=x^2-5x+6=(x-2)(x-3).$$

Here the roots are $x=2$ and $x=3$, so the graph crosses at two points.

Factorising and Solving Quadratic Equations

Factorising is one of the main methods for solving equations of the form $ax^2+bx+c=0$.

Example:

$$x^2-9x+20=0$$

Factorise:

$$(x-4)(x-5)=0.$$

Set each factor equal to zero:

$$x-4=0 \quad \text{or} \quad x-5=0.$$

So the solutions are

$$x=4 \quad \text{or} \quad x=5.$$

This method is especially useful when the quadratic factorises neatly. If it does not factorise easily, other methods such as completing the square or using the quadratic formula may be needed later in the course.

In IB Mathematics Analysis and Approaches SL, you should be able to recognise when factorising is the simplest and most efficient method. This is part of mathematical reasoning, not just routine calculation.

Common Mistakes to Avoid

When factorising quadratics, several errors can happen.

Forgetting to check for a common factor first.
Choosing two numbers that multiply correctly but do not add to the middle coefficient.
Mixing up signs, especially when one factor is negative.
Not checking the answer by expanding.

For example, if you factorise

$$x^2-2x-8,$$

you need two numbers that multiply to $-8$ and add to $-2$. The correct numbers are $-4$ and $2$, so

$$(x-4)(x+2).$$

A quick expansion check confirms:

$$(x-4)(x+2)=x^2-2x-8.$$

Checking is a strong habit in IB mathematics because it helps confirm accuracy and build confidence ✅.

Factorising in the Bigger Picture of Functions

Factorising quadratics is part of a wider set of skills in the Functions topic. Functions can be represented in multiple ways: as equations, tables, graphs, or descriptions. Factorised form is one important algebraic representation.

For quadratics, different forms give different information:

standard form $f(x)=ax^2+bx+c$ shows the coefficients clearly,
factorised form $f(x)=a(x-p)(x-q)$ shows the roots clearly,
vertex form $f(x)=a(x-h)^2+k$ shows the turning point clearly.

Understanding how these forms relate helps you move between representations. For example, if you know $f(x)=(x-2)(x+1)$, you can expand it to get $f(x)=x^2-x-2$. This helps when comparing models or using different solution methods.

This skill also supports composite and inverse functions later on, because algebraic manipulation is central to all function work. Even though factorising quadratics is a specific topic, it builds general fluency in working with expressions and equations.

Conclusion

Factorising quadratics is a key skill in IB Mathematics Analysis and Approaches SL because it connects algebra with function behaviour. It helps you solve equations, find roots, sketch graphs, and understand how different forms of a quadratic reveal different features. The factorised form $a(x-p)(x-q)$ is especially useful because it shows where the function is zero.

students, when you practise factorising, always remember to check for a common factor, choose the correct method, and verify your answer by expanding. Mastering this topic will make later work on graphs, equations, and function transformations much easier 🎯.

Study Notes

A quadratic function has the form $f(x)=ax^2+bx+c$, where $a\neq 0$.
Factorising means rewriting a quadratic as a product, often as $a(x-p)(x-q)$.
The roots or zeros are the values of $x$ that make $f(x)=0$.
If $f(x)=a(x-p)(x-q)$, then the roots are $x=p$ and $x=q$.
Always check for a common factor before factorising a quadratic.
For $x^2+bx+c$, find two numbers that multiply to $c$ and add to $b$.
For $ax^2+bx+c$ with $a\neq 1$, use split-middle-term or grouping methods.
Expanding is a reliable way to check whether factorising is correct.
Factorising helps with solving equations and sketching graphs.
In the Functions topic, factorised form gives important information about roots and graph behaviour.