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FACTORISING EQUATIONS

Factorising expressions with 2 terms There are only 2 ways of factorising a quadratic with two terms: 1. Look for something in common between the 2 terms. Often this will be an “x” or “a”

bracket (x+a) Eg: Factorise x2 + 3x ⇒ here, both terms have “x” =x(x+3) Eg: Factorise 3x(x-2) + 2(x-2) ⇒ here, the bracket (x-2) is common =(x-2)(3x+2) 2. The other case is where nothing is common. Usually, one term will be an “x2” and the

other term will be a number. This will be in the form a2-b2. Recall that the difference of perfect squares a2-b2 can be factorised into (a+b)(a-b)

Example 1: Factorise x2 – 9

You have to recognise that 9=32, and of course x2 is a perfect square. We now use the difference of perfect square formula to factorise into (a+b)(a-b)

∴ x2-9 = (x + 9 ) (x - 9 ) = (x+3) (x-3) Example 2: Factorise x2 – 17

Even though 17 isn’t a perfect square, we can still factorise it using the difference of perfect squares formula. We just don’t get a nice number in the brackets. x 2 – 17 = (x + 17 ) (x - 17 )

Example 3: Factorise 4px2 – 256p

This is a combination of the two methods. Here, 4 is a common factor and “p” is common too, so the first step is to take out “4p”. ⇒ 4p(x2 – 64) Now, recognise that inside the brackets we have a perfect square in the form a2 – b2. We now use the difference of perfect squares formula to factorise this further: ⇒ 4p(x + 8) (x – 8)



Factorising expressions with 3 terms These are expressions of the form ax2 + bx + c. When the co-efficient of x2 is 1, this is straight forward. Example 1: Factorise x2 + 3x + 2

Step 1: Find factors of the number (in this case the number is 2) The factors of 2 are “1” and “2”.

Step 2: Using the factors, try and add or subtract to make the co-efficient of “x”, which is 3 in this case. 2+1=3

Step 3: Open two pairs of brackets. ( ) ( ) Step 4: Write x as the first term in each bracket. (x ) (x ) Step 5: We wanted +2 and +1 to make 3, so we write them in the brackets in any

order. (x + 2) (x + 1) ∴x2 + 3x + 2 = (x + 2) (x + 1)

Example 2: Factorise a2-6a-7

Step 1: Factors of 7 are “1” and “7” Step 2: To make -6, we need -7 +1 Step 3: Open two brackets ( ) ( ) Step 4: Write x as the first term (x ) (x ) Step 5: Write in the -7 and the +1 (x-7) (x+1)

Example 3: Factorise t2-6t+8

Step 1: Factors of 8 are “4 and “2”, “8” and “1” Step2: To make -6, we need -4 and -2 (we can’t do anything with the 8 and 1 to

make -6) Step 3: ( ) ( ) Step 4: (x ) (x ) Step 5: (x – 4) (x – 2)

Continued next page



It becomes trickier when the co-efficient of x2 is not 1. Example 1: Factorise 2x2+5x+2

Step1: Here, we need to find factors of both the number “2” and the co-efficient of x2 (also 2 in this case).

Step 2: Now, take one of the factors on the left and multiply it by one on the right.

You do the same with the other pair.

We get 4 and 1 which we can add to make 5 (the co-efficient of x)

Step 3: Open the 2 brackets ( ) ( ) Step 4: Instead of writing x as the first term, we have to take into account the

factors we used. We split the “2” in front of the x2 into 2x1 so we write 2x and 1x

(2x ) (x ) Step 5: We paired the 2 with the other 2, and the 1 with the other 1. They go in

opposite brackets. (2x 1) (x 2) Step 6: The sign is whatever sign the factors of the number (2) were. In this case

they were both positive. (2x + 1) (x + 2)

2x2 + 5x + 2

2 2 1 1

2 x 2 = 4 1 x 1 = 1



Example 2: Factorise 15x2 + x – 2

Step 1: Factors:

Step 2: Trial and error 15 x 2=30 1 x 1=1 ⇒ can’t make 1 15 x 1 =15 1 x 2=2 ⇒ can’t make 1 5 x 2=10 3 x 1=3 ⇒ can’t make 1 5 x 1=5 3 x 2=6 ⇒ 6 – 5=1

So, we want the 5 and 3 to go with the 2 and 1. The 5 goes with the 1 and the 3 goes with the 2. (5x 2) (3x 1)

Step 3: We want to get positive 6 and negative 5. The 3x2 gave us 6, so both will

be positive. The 5x1 gave us 5 , so we need the 1 to be -1, so that we get -5 Step 4: Fill in the signs. (5x + 2) (3x – 1)

Completing the square In simple algebra, completing the square means to convert a quadratic equation. Example 1: Complete the square for x2 + 10x

Step 1: Take the co-efficient of x (10). Then divide it by 2 (5) and square it (25) Step 2: Add and subtract this number. X2 + 10x + 25 -25 Step 3: The first 3 terms will factorise to give a perfect square. X2 + 10 + 25 – 25 (x + 5) (x + 5) -25 (x + 5)2 – 25

15x2 + x - 2

15 1 2 1 5 3



Example 2: Complete the square for x2-4x-7

Step 1: Co-efficient of x is -4. Divide by 2 = -2. Square this figure and that will equal = 4.

Step 2: Add and subtract 4 x2 – 4x +4 -7 -4 (you write the +4 then the -4) Step 3: Factorise first three terms into a perfect square. x2 – 4x +4 -7 -4 = (x – 2) (x – 2) -11 = (x – 2)2 -11

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