Question

Factorise the following:

x² – 4y² – 2x + 4y

Answer 1

Given: x² – 4y² – 2x + 4y

Step-wise calculation:

1. Group terms to factor by grouping:

(x² – 2x) – (4y² – 4y)

2. Factor out common terms in each group:

x(x – 2) – 4y(y – 1)

3. Recognize that the expressions are close to forms involving terms like (x – 1)² or (y – 1)²; try completing the square for both x and y parts:

Rewrite the expression by rearranging terms:

x² – 2x – 4y² + 4y = (x² – 2x + 1) – (4y² – 4y + 1) – 1 + 1

Add and subtract 1 to complete square terms carefully:

(x – 1)² – [(2y)² – 2(2y)(1) + 1] + 0

Since (2y)² – 2 × 2y × 1 + 1 = (2y – 1)², this gives:

(x – 1)² – (2y – 1)²

4. The difference of squares factorization:

(a² – b²) = (a + b)(a – b)

Applying this:

(x – 1)² – (2y – 1)² = ((x – 1) + (2y – 1))((x – 1) – (2y – 1))

Simplify the factors:

(x – 1 + 2y – 1)(x – 1 – 2y + 1) = (x + 2y – 2)(x – 2y)