Study notes

Expansion & Factorisation

Expanding brackets with the special identities, factorising quadratic expressions, and simplifying algebraic fractions.

Learn it step by step

Squaring a sum: $(a+b)^2 = a^2 + 2ab + b^2$

Expanding $(a + b)^{2}$ means multiplying $(a + b)$ by itself. Every term in the first bracket must multiply every term in the second, giving $a^{2} + ab + ab + b^{2} = a^{2} + 2 ab + b^{2}$ . The middle term is always $2 \times$ the two letters/numbers multiplied together.

One row of the expansion of

(x + 4)^{2}

: the

x

row gives

x^{2}

and

4 x

; the

+ 4

row (not shown) gives another

4 x

and

16

, so the full expansion is

x^{2} + 8 x + 16

Squaring a difference: $(a-b)^2 = a^2 - 2ab + b^2$

The same idea applies when one term is negative: $(a - b)^{2} = (a - b) (a - b) = a^{2} - ab - ab + b^{2} = a^{2} - 2 ab + b^{2}$ . The middle term is negative because a negative times a positive gives a negative, but the last term $b^{2}$ is still positive because a negative times a negative is positive. Forgetting the middle term entirely (writing $a^{2} - b^{2}$ instead) is the single most common Sec 2 slip.

Difference of two squares: $a^2 - b^2 = (a+b)(a-b)$

When two brackets differ only by the sign in the middle, like $(a + b) (a - b)$ , the cross terms $- ab$ and $+ ab$ cancel out, leaving only $a^{2} - b^{2}$ . This works in reverse too: any expression written as one square minus another square, such as $x^{2} - 25$ , factorises straight to $(x + 5) (x - 5)$ .

x^{2} - 25 = (x + 5) (x - 5)

is zero exactly at

x = - 5

and

x = 5

, the two numbers being squared.

Factorising $x^2 + bx + c$: find two numbers that multiply to $c$ and add to $b$

For a quadratic with leading coefficient 1, look for two numbers whose product is the constant term and whose sum is the coefficient of $x$ . For $x^{2} + 7 x + 12$ , the numbers 3 and 4 multiply to 12 and add to 7, so $x^{2} + 7 x + 12 = (x + 3) (x + 4)$ . Always check by expanding your answer back out.

Factorising a non-monic quadratic $ax^2+bx+c$ ($a \neq 1$)

When the coefficient of $x^{2}$ is not 1, you must find a pair of brackets $(p x + q) (r x + s)$ where $p r = a$ , $q s = c$ , and the cross terms add to give $b$ . For $2 x^{2} + 7 x + 3$ , try $(2 x + 1) (x + 3)$ : expanding gives $2 x^{2} + 6 x + x + 3 = 2 x^{2} + 7 x + 3$ , which matches. Always expand your trial factorisation to check the middle term before finalising it.

Simplifying an algebraic fraction means factorising first, then cancelling

You can only cancel a factor that is common to the WHOLE numerator and the WHOLE denominator, never a term buried inside a sum. For $\frac{x ^{2} - 9}{x ^{2} + 5 x + 6}$ , factorise both: $\frac{( x + 3 ) ( x - 3 )}{( x + 2 ) ( x + 3 )}$ , then cancel the common bracket $(x + 3)$ to leave $\frac{x - 3}{x + 2}$ .

Worked examples

Expand and simplify:

(x + 5)^{2}

Use $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ with $a = x$ , $b = 5$ .
$x^{2} + 2 (x) (5) + 5^{2}$
$= x^{2} + 10 x + 25$ .

Factorise fully:

3 x^{2} + 11 x + 6

The leading coefficient is 3, so try brackets of the form $(3 x + q) (x + s)$ or $(3 x + s) (x + q)$ .
Try $(3 x + 2) (x + 3)$ : expanding gives $3 x^{2} + 9 x + 2 x + 6 = 3 x^{2} + 11 x + 6$ , which matches.
Answer: $(3 x + 2) (x + 3)$ .

Simplify the algebraic fraction:

\frac{4 x ^{2} - 1}{2 x ^{2} + 5 x - 3}

Factorise the numerator using the difference of two squares: $4 x^{2} - 1 = (2 x + 1) (2 x - 1)$ .
Factorise the denominator: need $(2 x + q) (x + s)$ with $q s = - 3$ and cross terms giving $5 x$ ; try $(2 x - 1) (x + 3)$ : $2 x^{2} + 6 x - x - 3 = 2 x^{2} + 5 x - 3$ , which matches.
Cancel the common factor $(2 x - 1)$ : $\frac{( 2 x + 1 ) ( 2 x - 1 )}{( 2 x - 1 ) ( x + 3 )} = \frac{2 x + 1}{x + 3}$ .

A square photo frame has outer side

(x + 3)

cm. A square photo of side

(x - 1)

cm sits inside it, centred, leaving an even border. Find, in expanded and simplified form, the area of the border (the frame minus the photo).

Outer area: $(x + 3)^{2} = x^{2} + 6 x + 9$ .
Inner (photo) area: $(x - 1)^{2} = x^{2} - 2 x + 1$ .
Border area = outer minus inner: $(x^{2} + 6 x + 9) - (x^{2} - 2 x + 1) = 6 x + 9 + 2 x - 1 = 8 x + 8$ .

Mind map

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Expansion & Factorisation

Learn it step by step

Squaring a sum: \((a+b)^2 = a^2 + 2ab + b^2\)

Squaring a difference: \((a-b)^2 = a^2 - 2ab + b^2\)

Difference of two squares: \(a^2 - b^2 = (a+b)(a-b)\)

Factorising \(x^2 + bx + c\): find two numbers that multiply to \(c\) and add to \(b\)

Factorising a non-monic quadratic \(ax^2+bx+c\) (\(a \neq 1\))

Simplifying an algebraic fraction means factorising first, then cancelling

Worked examples

Mind map