GCSE Factorising revision

A quick, one-off masterclass in how to put things into brackets today - six methods of factorising you need to know to do well at GCSE maths.

(1) Common number

$3a+6$

• two terms (letter and number, no squares)

• you can divide them both by 3

• $3\times a=3a$

• $3\times 2=6$

• $3a+b=3(a+2)$

Try these:

$9b+12$ $12c-8$ $90d+15$ $7e-14$

(2) Common letter

$x^2-4x$

• two terms (letter-squared and letter)

• you can divide them both by $x$

• $x\times x=x^2$

• $x\times -4=-4x$

• $x^2-4x=x(x-4)$

Try these:

$b^2+9b$ $c^2-2c$ $d^2+10d$

(3) Common number and letter

$6x^2-4x$

• two terms (letter-squared and letter)

• you can divide them both by $2x$

• $2x\times 3x=6x^2$

• $2x\times -2=-4x$

• $6x^2-4x=2x(3x-2)$

Try these:

$2x^2+8x$ $3x^2-6x$ $4x^2-10x$

(4) Difference of two squares

$4x^2-25$

• two terms (both squares)

• Use $(a+b)(a-b)=a^2-b^2$

• $4x^2=(2x{)}^2$, so $a=2x$

• $25=5^2$, so $b=5$

• $4x^2-25=(2x-5)(2x+5)$

Try these:

$x^2-9$ $9x^2-1$ $4a^2-9b^2$

(5) Regular quadratic

$x^2-2x-15$

• Three terms (letter-squared, letter and number)

• Look for two numbers $p$ and $q$ such that:

• $pq=-15$; and

• $p+q=-2$. (Adding things in the middle would be the end of the Times!)

• Not too many possibilities that multiply to -15: 1 and -15, 3 and -5, 5 and -3, 15 and -1. Only 3 and -5 work.
• $x^2-2x-15=(x+3)(x-5)$

Try these:

$x^2+3x+2$ $x^2-3x+2$ $x^2+13x+36$ $x^2+2x-35$

(6) Quadratic with a number in front

$4x^2+8x+3$

• Three terms (letter-squared, letter and number)
• More difficult! Magic number is $4\times 3=12$

• Want two numbers $p$ and $q$ such that:

• $pq=12$

• $p+q=8$

• 2 and 6 work!
• Split up $8x$ as $2x+6x$ and write out: $4x^2+2x+6x+3$
• Factorise first half: $2x(2x+1)$
• Factorise second half: $3(2x+1)$
• Combine: $(2x+3)(2x+1)$ - phew!

Try these:

* $2x^2+3x+1$ * $3y^2+8y-3$ * $4z^2+5z+1$

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* Edited 2016-05-08 to correct wrong letters in last two questions. Thanks, Rosie, for pointing out my error.