Factorising reverses the special identities: a2+2ab+b2=(a+b)2, a2-2ab+b2=(a-b)2 and a2-b2=(a+b)(a-b). Spotting them is quicker than using a frame. Why: x2+6x+9 is made of x2, two strips of 3x and 3 x 3=9. These pieces fit together as a square of side x+3, so x2+6x+9=(x+3)2. A perfect square has square end terms and a middle term equal to twice the product of their square roots. **Pattern** | **What to look for** | **Example** a2+2ab+b2 | square end terms, middle term +2ab | 4x2+12x+9=(2x+3)2 a2-2ab+b2 | square end terms, middle term -2ab | 25y2-10y+1=(5y-1)2 a2-b2 | two squares with a minus sign between them | 49x2-16=(7x+4)(7x-4) Method - Take out any common factor first. - Two terms that are squares with a minus sign between them: use a2-b2=(a+b)(a-b). - Three terms with square end terms: name a and b and check that the middle term is 2ab; then write (a+b)2 or (a-b)2. - A sum of two squares, such as x2+9, does not factorise. Fractions follow the same checks. In x2+3x+ 94, the end terms are x2 and (32)2, and the middle term is 2 x x x 32=3x. So x2+3x+ 94=(x+ 32)2. Check: (x+ 32)2=x2+2(32)x+ 94=x2+3x+ 94. Example Factorise completely 3x2-75 and 9a2-30ab+25b2. 3x2-75 = 3(x2-25) = 3(x+5)(x-5) 9a2-30ab+25b2 = (3a)2-2(3a)(5b)+(5b)2 = (3a-5b)2 Check: expand (3a-5b)2=9a2-30ab+25b2. A common error is x2-36=(x-6)2: the square of a difference has a middle term, -12x, so x2-36 is a difference of two squares instead.

Factorising with special identities

Secondary 2

Factorising reverses the special identities: $a^2+2ab+b^2=(a+b)^2$, $a^2-2ab+b^2=(a-b)^2$ and $a^2-b^2=(a+b)(a-b)$. Spotting them is quicker than using a frame.

Why: $x^2+6x+9$ is made of $x^2$, two strips of $3x$ and $3\times3=9$. These pieces fit together as a square of side $x+3$, so $x^2+6x+9=(x+3)^2$. A perfect square has square end terms and a middle term equal to twice the product of their square roots.

Pattern	What to look for	Example
$a^2+2ab+b^2$	square end terms, middle term $+2ab$	$4x^2+12x+9=(2x+3)^2$
$a^2-2ab+b^2$	square end terms, middle term $-2ab$	$25y^2-10y+1=(5y-1)^2$
$a^2-b^2$	two squares with a minus sign between them	$49x^2-16=(7x+4)(7x-4)$

Method

Take out any common factor first.
Two terms that are squares with a minus sign between them: use $a^2-b^2=(a+b)(a-b)$.
Three terms with square end terms: name $a$ and $b$ and check that the middle term is $2ab$; then write $(a+b)^2$ or $(a-b)^2$.
A sum of two squares, such as $x^2+9$, does not factorise.

Fractions follow the same checks. In $x^2+3x+\tfrac{9}{4}$, the end terms are $x^2$ and $\left(\tfrac{3}{2}\right)^2$, and the middle term is $2\times x\times\tfrac{3}{2}=3x$. So $x^2+3x+\tfrac{9}{4}=\left(x+\tfrac{3}{2}\right)^2$. Check: $\left(x+\tfrac{3}{2}\right)^2=x^2+2\left(\tfrac{3}{2}\right)x+\tfrac{9}{4}=x^2+3x+\tfrac{9}{4}$.

Example

Factorise completely $3x^2-75$ and $9a^2-30ab+25b^2$.

$\begin{aligned} 3x^2-75 &= 3(x^2-25) \\ &= 3(x+5)(x-5) \end{aligned}$

$\begin{aligned} 9a^2-30ab+25b^2 &= (3a)^2-2(3a)(5b)+(5b)^2 \\ &= (3a-5b)^2 \end{aligned}$

Check: expand $(3a-5b)^2=9a^2-30ab+25b^2$. A common error is $x^2-36=(x-6)^2$: the square of a difference has a middle term, $-12x$, so $x^2-36$ is a difference of two squares instead.

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Pattern	What to look for	Example
\(a^2+2ab+b^2\)	square end terms, middle term \(+2ab\)	\(4x^2+12x+9=(2x+3)^2\)
\(a^2-2ab+b^2\)	square end terms, middle term \(-2ab\)	\(25y^2-10y+1=(5y-1)^2\)
\(a^2-b^2\)	two squares with a minus sign between them	\(49x^2-16=(7x+4)(7x-4)\)