Expanding and Factorising

Expanding Single Brackets

To expand a single bracket, multiply each term inside the bracket by the term outside.

$3(x + 4) = 3x + 12$

$-2(3x - 5) = -6x + 10$

Notice in the second example: $-2 \times (-5) = +10$. Be careful with negative signs.

More complex examples:

$x(x + 7) = x^2 + 7x$

$-4x(2x - 3) = -8x^2 + 12x$

Expanding Double Brackets

To expand two brackets, multiply every term in the first bracket by every term in the second bracket. A common method is FOIL (First, Outer, Inner, Last).

$(x + 3)(x + 5)$

• First: $x \times x = x^2$
• Outer: $x \times 5 = 5x$
• Inner: $3 \times x = 3x$
• Last: $3 \times 5 = 15$

$= x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

Expanding with Negative Terms

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

$= x^2 - 2x + 4x - 8 = x^2 + 2x - 8$

$(x - 3)(x - 7)$

$= x^2 - 7x - 3x + 21 = x^2 - 10x + 21$

Remember: negative × negative = positive.

Expanding $(ax + b)(cx + d)$

When the coefficient of $x$ is not 1:

$(2x + 3)(3x - 1) = 6x^2 - 2x + 9x - 3 = 6x^2 + 7x - 3$

Squaring a Bracket

$(x + 4)^2 = (x + 4)(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$

A common error is to write $(x + 4)^2 = x^2 + 16$. This is wrong — you must include the middle term.

The general pattern is:

$(a + b)^2 = a^2 + 2ab + b^2$

$(a - b)^2 = a^2 - 2ab + b^2$

Difference of Two Squares (DOTS)

When you expand $(a + b)(a - b)$:

$(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2$

The middle terms cancel. This is the difference of two squares.

Example: $(x + 6)(x - 6) = x^2 - 36$

Factorising — Taking Out a Common Factor

Factorising is the reverse of expanding. Look for the highest common factor (HCF) of all terms.

$6x + 15 = 3(2x + 5)$

$8x^2 - 12x = 4x(2x - 3)$

Always check: if you expanded your answer, would you get back to the original expression?

Factorising Quadratics ($x^2 + bx + c$)

To factorise $x^2 + bx + c$, find two numbers that:

• Multiply to give $c$
• Add to give $b$

Example: Factorise $x^2 + 7x + 12$

Find two numbers that multiply to 12 and add to 7: $3$ and $4$.

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

Example: Factorise $x^2 - 2x - 15$

Find two numbers that multiply to $-15$ and add to $-2$: $-5$ and $3$.

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

Factorising the Difference of Two Squares

Recognise the pattern $a^2 - b^2 = (a + b)(a - b)$.

$x^2 - 49 = (x + 7)(x - 7)$

$4x^2 - 25 = (2x + 5)(2x - 5)$

Factorising Harder Quadratics ($ax^2 + bx + c$ where $a \neq 1$)

For quadratics where the coefficient of $x^2$ is not 1, use the AC method:

1. Multiply $a \times c$
2. Find two numbers that multiply to $ac$ and add to $b$
3. Split the middle term and factorise in pairs

Example: Factorise $2x^2 + 7x + 3$

$a \times c = 2 \times 3 = 6$. Find two numbers that multiply to 6 and add to 7: $6$ and $1$.

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

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

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

Tip 1: Always Expand to Check Your Factorisation

After factorising, expand your brackets to verify you get the original expression. This takes seconds and catches errors.

Tip 2: Look for Common Factors First

Before attempting to factorise a quadratic, check whether all terms share a common factor. For example, $2x^2 + 10x + 12 = 2(x^2 + 5x + 6) = 2(x + 2)(x + 3)$.

Tip 3: Remember the Sign Rules

When factorising $x^2 + bx + c$:

• If $c$ is positive, both signs are the same (both $+$ or both $-$, matching the sign of $b$)
• If $c$ is negative, the signs are different

Tip 4: Spot DOTS Immediately

If you see $x^2 - \text{number}$ with no $x$ term, check if the number is a perfect square. If so, it is a difference of two squares.

Tip 5: Use the Grid Method if FOIL Confuses You

Draw a 2×2 grid. Place terms from each bracket along the top and side. Multiply to fill in the grid, then collect like terms.