Question

Factorise by substituting terms:

3(x – y)² – 4x + 4y – 4

Answer 1

We are given the expression:

3(x – y)² – 4x + 4y – 4

Step 1: Use substitution.

Let a = x – y.

Then the expression becomes:

3a² – 4x + 4y – 4

Now substitute back x = a + y to express everything in terms of a and y:

–4x + 4y

= –4(a + y) + 4y

= –4a – 4y + 4y

= –4a

So the expression becomes:

3a² – 4a – 4

Step 2: Factorise 3a² – 4a – 4.

We look for two numbers that multiply to 3 × (–4) = –12 and add up to –4: these are –6 and 2:

3a² – 6a + 2a – 4

= 3a(a – 2) + 2(a – 2)

= (3a + 2) (a – 2)

Step 3: Replace a = x – y:

[3(x – y) + 2] [(x – y) – 2]

= (3x – 3y + 2) (x – y – 2)