Expansion and Factorisation

When two brackets are multiplied together, like $(x + 4)(x + 2)$, you multiply every term in the first by every term in the second. That gives four products, and the two middle ones usually collect together.

For $(x + 4)(x + 2)$ the four products are $x^2$, $2x$, $4x$ and $8$. Adding them: $x^2 + 2x + 4x + 8 = x^2 + 6x + 8.$

The two middle terms $2x$ and $4x$ are like terms, so they collect to $6x$.

A few more, watching the signs:

• $(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15$.
• $(x + 2)^2 = (x + 2)(x + 2) = x^2 + 4x + 4$.

A grid keeps you sure you have caught all four products. Multiply, fill the grid, then collect the like middle terms.

When you expand two brackets like $(x + 4)(x + 2)$, the answer $x^2 + 6x + 8$ has a name: it is a quadratic expression. The clue is the $x^2$ term — that highest power of 2 is what makes it quadratic.

A quadratic of this kind always has three parts: an $x^2$ term, an $x$ term, and a number.

We describe this shape as $x^2 + bx + c$, where:

• $b$ is the coefficient of $x$ — here $b = 6$.
• $c$ is the number term — here $c = 8$.

Why does this matter? Because expanding and factorising are opposites. You have just seen $(x + 4)(x + 2)$ open up into $x^2 + 6x + 8$. Soon you will go the other way — taking a quadratic and finding the two brackets it came from. Spotting the form $x^2 + bx + c$ is the first step of that journey.

Factorising is the reverse of expanding — it puts brackets back. The simplest kind takes out a common factor: a number or letter that divides into every term.

To factorise, follow three steps:

1. Find the highest common factor (HCF) of every term.
2. Write the HCF outside a bracket.
3. Divide each term by the HCF — the results go inside.

Example: Factorise $12x^2 + 8x$. The number HCF of 12 and 8 is 4, and both terms have an $x$, so the HCF is $4x$. Dividing, $12x^2 \div 4x = 3x$ and $8x \div 4x = 2$, giving $4x(3x + 2)$.

Always take out the highest factor. $12x^2 + 8x = 2(6x^2 + 4x)$ is partly factorised — but $4x(3x + 2)$ is fully done.

This skill matters here because not every quadratic factorises into two brackets — some just have a common factor. Always check for one first.

Here is the big Stage 9 skill: factorising a quadratic of the form $x^2 + bx + c$ back into two brackets.

The method is a number hunt. You need two numbers that:

• add together to make $b$ (the coefficient of $x$), and
• multiply together to make $c$ (the number term).

Example: Factorise $x^2 + 6x + 8$. You need two numbers that add to 6 and multiply to 8. Try the factor pairs of 8: $1 \times 8$ (adds to 9, no) and $2 \times 4$ (adds to 6, yes!). So the two numbers are 2 and 4, and: $x^2 + 6x + 8 = (x + 4)(x + 2).$

The two numbers drop straight into the brackets. Always check by expanding your answer — if it returns to the original quadratic, you are right.

Signs are where most slips happen. Reading the signs of $b$ and $c$ tells you what kind of two numbers to hunt for.

Two cases cover almost everything:

• $c$ positive: the two numbers have the same sign, matching the sign of $b$. For $x^2 - 7x + 12$, you need two negatives that multiply to $+12$ and add to $-7$ — that is $-3$ and $-4$, giving $(x - 3)(x - 4)$.
• $c$ negative: the two numbers have opposite signs. For $x^2 + 2x - 15$, you need a positive and a negative multiplying to $-15$ and adding to $+2$ — that is $+5$ and $-3$, giving $(x + 5)(x - 3)$.

Work calmly: list the factor pairs of $|c|$, then decide the signs from these two rules. Test each candidate by adding — and always expand at the end to be sure.

Getting confident with expanding and factorising quadratics now pays off enormously:

• Solving quadratic equations at IGCSE relies directly on factorising into two brackets.
• Quadratic graphs — the parabola — are sketched using the factorised form to find where the curve crosses the axis.
• Algebraic fractions are simplified by factorising the top and bottom, then cancelling.
• Areas model expansion — a rectangle $(x + 4)$ by $(x + 2)$ has area $x^2 + 6x + 8$.

In everyday life, factorising is spotting hidden structure — seeing that a tricky-looking total is really two simple amounts multiplied together.

If quadratic work feels hard later, it is usually a sign slip in the two-number hunt underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.