Factorising Quadratics

A quadratic expression has an x^2 term and no higher power of x, such as x^2+8x+15

To factorise means to write an expression as a product: things multiplied together. A quadratic like x^2+8x+15 usually factorises into two brackets:

x^2+8x+15=(x+3)(x+5)

Factorising is expanding backwards. If you expand (x+3)(x+5) you get x^2+8x+15 again, so you can always check your answer by expanding it.

In the exam you are usually asked to factorise an expression such as x^2+6x+8, for 2 marks. Sometimes it is a difference of two squares, such as x^2-36, for 1 mark. These questions are usually near the end of the paper, often straight after a part that asks you to expand two brackets. There is no equals sign: you are not solving anything.

Expand (x+3)(x+5) with a grid. The four parts are x^2, 5x, 3x and 15

Together that is x^2+5x+3x+15=x^2+8x+15. Look where the 8 and the 15 come from: 3+5=8 and 3\times5=15

This always happens. If the numbers in the brackets are p and q:

(x+p)(x+q)=x^2+(p+q)x+pq

The two numbers add to give the number in front of x, and multiply to give the number on its own.

So to factorise x^2+bx+c, find two numbers that multiply to give c and add to give b. Start with the multiplying: list the factor pairs of c, then pick the pair that adds to b

The rule is the same: multiply to give the number on its own, add to give the number in front of x. The signs tell you which signs the two numbers have.

• Number on its own positive, x term positive: both numbers are positive.
• Number on its own positive, x term negative: both numbers are negative, because negative times negative is positive.
• Number on its own negative: one number is positive and one is negative. The bigger one (ignoring its sign) has the same sign as the x term.

For example, x^2-8x+15 needs two numbers that multiply to 15 and add to -8. They are -3 and -5, so x^2-8x+15=(x-3)(x-5)

Sometimes both numbers are the same. x^2+10x+25 needs two numbers that multiply to 25 and add to 10: they are 5 and 5. So x^2+10x+25=(x+5)(x+5), which can be written (x+5)^2

A difference of two squares is one square subtracted from another, such as x^2-81. There is no x term, so the two numbers must add to 0 and multiply to -81: they are 9 and -9

x^2-81=(x+9)(x-9)

In general, x^2-k^2=(x+k)(x-k), where k is a whole number. Expanding shows why: the +kx and the -kx cancel out. It is worth knowing the square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

If there is no number on its own, x is a common factor, and you need only one bracket. For example, x^2-6x=x(x-6). Don't try to make two brackets.

So before you start, look for a number on its own. x^2-6x+8 has one, 8, so it needs two brackets: (x-2)(x-4). x^2-6x has none, so it needs one bracket: x(x-6)

• Taking out x from only two of the terms. x(x+6)+8 is not factorised; a quadratic with a number on its own needs two brackets.
• Choosing numbers that multiply correctly but don't add correctly. (x+1)(x-25) is not x^2-25; the pair must also add to the number in front of x, so it is (x+5)(x-5)
• Getting the signs the wrong way round. For x^2+2x-8 the bigger number takes the sign of the x term: (x+4)(x-2), not (x-4)(x+2)
• Putting the x term inside a bracket. (x+7x)(x+1) is not x^2+8x+7; each bracket holds x and a number, so it is (x+1)(x+7)
• Using two brackets when there is no number on its own. x^2-5x is x(x-5), with one bracket.
• Trying to solve it. There is no equals sign, so don't give values of x; write the expression as brackets and check by expanding.